Don't the boys have two bio parents? You all have custody of the boys? Does their bio-mom pay child support or contribute? I would think that both bio parents contributing would suffice, but I must be missing something here. Have you all sat down and budgeted kid costs? Are there other areas in your life where you could cut expenses? 
For your husband to answer that you all should move is drastic and obviously not a real solutions-oriented response. 
 

We/I suffered financially (unplanned) after we married because of our poor planning and the surprise demands from the skids/their mother . . . and DH having loose backbone issues at the time. I carried an unfair burden for some years, but DH promised to make it up to me, and he has, so it has all evened out. Otherwise, I would probably look like a bar-b-q cheese crisp from resentment. 

So I prepared for my daughter before I had her (finished school first then looked to start a family). I work in the medical field also (NP) and when I met DH I was just graduating from RN school so I made double what he made. He owned his apartment so I only gave him half of the maintenance fee, and half of the light bill. I normally ordered takeout and sometimes I would fund trips so we could all experience things together. 

When DD was born, I wanted to move to a better area, so we saved up and got a house. I, too paid the lion's share of the bills, because DH could only afford to help with utilties/food/1/3 of mortgage after his child support for youngest SS and he had custody of his first 2 sons, but their mom barely paid CS. I was okay with having to shoulder more costs because the house was big and the area was nice so I thought it was worth it. I did envision that his kids would grow up and stay with us until they were ready to leave (I wanted them to go to college, or work and pay the light bill but also save to buy their own house, but that didn't pan out). I did focus on DD more than his kids.  I felt that I worked and planned for DD and that was my approach to parenting. DH had sex and his kids were a byproduct (DD was the only planned one, only because of me). I didn't see it as my fault he couldn't afford to pay for activities for his kids (and I still don't). I think it's sad, and like you, I do want the best for all of my stepsons. But, I am not their mom, I am only obligated to DD. So yes, DD is in a private school, and after school activities, and I am fortunate that I am able to easily pay for it myself. The rest of my surplus, I invest, or buy presents for my friends and family (and sometimes DH's family).  

When I met DH I knew he made a lot less than I did, but I did expect that his contribution to parenting DD be fair. So currently, he helps as much as he can because he isn't working. This means, he does the pickups, after school activities, the cooking, laundry and cleaning. When he starts to work, I do expect for him to pay for what he can within reason. 

You guys just have to work out what feels fair to pay for each of you to contribute to household expenses (mortgage, light, food).  Does he have custody of his kids? If so, you may have to tweak food costs (teen boys oftentimes eat more than little girls). After the household bills are considered, I would keep my surplus and DH does whatever with his (though he never really had a surplus). 

It is NOT terrible that you are successful and are able to help your girls, you are NOT obligated to do for anyone else. Your stepsons have a mom and a dad. Your daughters have you (and maybe biodad?). That's not on you to strive to want to do after schools and the extras for your stepsons, that's on your DH and their own mom. If he (or their mom) wants the extras for his kids, they should work for it (just like you and I do). As it relates to  my stepsons, I only do what I am comfortable doing (which these days is not much). Hope this helps!

edit to add: we have always kept separate accounts. When we lived in DH's apartment, I gave him half of the bills (which was only about 600 monthly) and I kept my surplus. I as able to save so much because DH owned his apartment so we never paid market value rents. When we moved to the house, I would have him send me 80% of his check for our bills (which only covered about 1/4 of our bills).

When we try to discuss this with him he argues that our expenses are too high and we should move.

BTW, who is we and who is him?  Just so I can be clear.

I call bullshit on his miscalculating what he can contribute.  It sounds far more like he manipulated his way into a marriage with a sugar mama.

If I were you, I would continually rub his nose in the $600/mo he owes to the marital income. Stop working to cover his lies and manipulations.  Provide the things for your DD's that you can, he can figure out how to make up the $600/mo and provide those things for his DSs that you provide for your DDs..  

No guilt. All that does is have you torturing yourself while he is falling short, not working the extra job to deliver on his commitment as your equity life partner, or to provide the things for his boys that he laments you providing for your girls. Do not sacrifice the benefits you provide for your girls because he is a failure as an earner, failure as a partner, and failure as a father.

IMHO of course.

Our blended family marriage is a bit different and simple by comparison.  We married right after I graduated with my undergrad.  She was a rull time studen/single mom. SS was 15mos old when we met. We married a week before his 2nd B-day.

We married with nothing but my newly printed degree, two 8yo cars, and 2 apartments worth of college furniture.  DW was a SAHM for the first 3yrs we were married. Which was just fine.  When SS-31 started kindergarten she went to work while remaining a full time student.

So, what we have we have built together.  SS remained an only child in our marriage so we did not have the yours and mine or ours dynamic to deal with.

Since we married DW has completed a dual major BS, we both have completed a Masters each, and a professional certification each.  We both have lucrative professional careers.   We have no my income or her income dynamic. It is all marital income. That has always been the case even when I was the sole earner.

Neither of us has failed to step up to support our marriage and family.

Don't tolerate him sandbagging on being an equity partner or in contributing as planned to the marital coffers.

All IMHO of course.

I trusted that he had correctly figured out how much he could contribute... is now paying $600 a month less

I hope you sat with him while he line itemed his expenses before you accepted that he could only contribute $600 less. 

Spouse income disparity is common. Ours is even bigger now that we are retired. Mine is twice his. I pay 2/3 of joint bills vs his 1/3. We have adult kids. He gifts his kids (and grands). I do mine. Do mine get more than his? Yes. Do I feel badly? No. If we were not married, his kids would get even less as more of his income would be spent on necessities. 

His kids have two parents. Your kids have you. All kids should be supported by their bio parents. Can you occasionally buy his kids something? Sure. But it should not be an expectation.

" If I'm willing to work extra to make it happen for them, maybe he should do the same?  "

Absolutely! Actually, you shouldn't be taking on extra work to cover HIS expenses at all. He is expecting you to pay for a large share of his part of the mortgage AND his sons' expenses. Nope. Not fair. Time for him to step up and either get a second job or a better paid job. His sons are his responsibility. If your daughters get things that they don't, they take it up with their father (or their mother - is she in the picture at all?). Are the boys with you full time?

I'm also inclined to think that he deliberately misled you about his income because he just expected  you to pick up the slack. Maybe he even thought that you wouldn't notice ...

He could not be off that much.  Not knowing of a $600 expense. OK less look at this differently.  600/4 is 150 a week.  Walmart pays $12 a hour. He was to work at Walmart 20 hours a week. $ 240 a week [ - taxes, other thing] . You will have exter money. 
He has three kids, they need someplace to sleep.  You can't move into a trailer. He has to pull his own weight. 
'Maybe this was one of the courses for his first marriage to fail,  Nothing is more sexy  then a lazy man who lets his wife support him.

I had to do this when married to my big-spender ex. He had kids, I don't. He HATED this as it ended his ability to spend alllll his $$ plus my earnings. It worked very well for me, especially the part where there were no more overdrafts on any account that had my name & credit score attached.

Direct-deposit into my/his respective checking accounts, with a 3rd "joint account" we each transferred the same amount of $$ into for shared monthly bills like rent, utilities, insurance, etc. That way, xH spending on his kids was limited to funds he had AFTER our bills were paid.

Unsurprisingly, the gifts his kids got became MUCH less extravagant and far less frequent. XH loved spending Other People's Money (specifically MY earnings) on his kids.

Boundaries matter!

Like StepUltimate, we had to separate accounts to rein in DH's spending on SD62's never-ending problems and wants.  It works.

Another thought from a 79-year old.  Conditions change over time.  When we got together, DH was much the higher earner.  Then I became a SAHM so he was the only earner.  He uncomplainingly supported my 2 bios since their dad was a deadbeat.  After 10 years, I went back to work and our earnings were roughly equal for about 10 years.  When he lost his job, I was the sole support for 3 or 4 years.  He was employed again and our earnings were about equal though my employer offered a generous 401K and his did not.  The kids were gone by then so we super-maxed the 401K.

Nowadays, the 401K supplements our Social Security and it would be easy for me to think,  "I'm supporting him" since most of his"allowance" goes to supplement SD62's living expense.  But none of that would have been possible without his early earnings, support of my SAHM years and support of my bios.

So, over time, things change and you might see the same over the length of your marriage.  Good luck.

Does your husband understand the concept of transaction costs?  It's easy to say we should move, but selling a house, buying a new one, getting a new mortgage and physically moving a household is very expensive. Is HE going to shoulder that cost and the work that would go into relocating? Yeah, I didn't think so, but it was his failure to plan and budget that caused this issue. 

I would continue to do for my own bios and let your husband and the boys' mother provide for them.  It's not your job - those kids have 2 parents.  If your husband complains that you should be subsidizing his kids, I'd ask him straight out of he married you for your money/earning potential. Your kids should not do without things you want to do for them because he can't do the same for his sons.  I'm not opposed to the higher earner picking up a higher percentage of household costs, but that doesn't necessarily extend to kids' extras.  Also, if you can take on extra work to pick up his slack, so can he.  He seems more inclined to have you pay for him and his than to find ways to pay for himself.  If your finances aren't already separate, I'd separate them. 

I think with blended situations where both sides bring children, its a difficult balance. There is just a natural pull away from the traditional nuclear family value of all kids equal spending/ equal opportunities.

For example... they have (2) different primary parents then their step siblings. Entirely different grandparents and extended family also. This often brings significant imbalance in regards to incomes, wealth, opportunities etc etc

For couples... it is really hard because you are sharing some part of the financial burden, while also maintaining financial independence.  You are actually already paying a fair amount to support his children by covering a higher portion of the monthly family expenses.  In addition, you are then further subsidizing that based on him making an error or whatever it was?  

I don't think it makes any sense for you to limit your own childrens opportunities, and I think it would probably be good without being confrontational about it, that you are in fact already subsidizing him and his children by paying a much larger share of the fixed household expenses...

I'm curious if anyone has real world expereince of how this plays out later on down the line?  Do you envision combing finances when kids are out of the picture?  If not, how do you feel about providing far, far more towards retirement goals then your spouse?  With bigger picture stuff like college, or trade school- how do step siblings get along if one has that paid for, and the other(s) don't?  Tricky stuff! 

I find this board to be a much needed source of perspective and understanding too.

My dh and I signed a prenup before we got married and one stipluation was that neither would be financially responsible for the other one's child. I think that's only fair. My BS has special needs, but I didn't expect a prospective partner to take on the extra expense involved in that. The rate of divorce in second marriages are very, very high. If things don't work out for you and your husband, would you still be okay with diverting money to his kids that you could have invested in your daughters or saved for the future?

And separate spending. 

No guilt in providing for your kiddos when he cannot provide for his.