Give her a month (maybe two) and let her know that it's over. 

I see some pretty big red flags here obviously.

1- She separated and jumped into a relationship with you before the ink was dry. She "can't" be on her own or not in a relationship. 

2- As a step parent to an adult stepson on the spectrum, that is a LOT to take on. Some days are better than others. I knew from the start that my SS20 would not likely ever live on his own. However, DH is a HUGE supporter of me and I know he has my back 100% of the time to help with SS's behavior issues. Are you sure you have that kind of support? Does she have a long term plan for them? Where on the spectrum do the 2 fall? 5 yr old in diapers is very concerning. Let me tell you, there is nothing like a 19 yr old shitting his pants because he doesn't like having bowel movements and finally his body takes care of it for him. This is my life- are you sure you want in? 

3- Is her ADHD son medicated? My BS has ADD, he is a borderline genius, he is sweet and kind, however- I can tell in 2 minutes flat of walking in the door if he took his meds that day. He is like ricochet rabbit in his thoughts. We always joke that he is going to cure cancer one day but forget his pants. Meds make a difference. Not just for the rest of the family but for him too. He is a better brother and friend when he is on the meds- impulse control is a situation with kids who have ADHD. 

I don't know if you should end your relationship or not, only you know that. If you do end it, like Winterglow said- give her 30-60 days to figure it out. Then she and ker kids need to move out. 

Have you talked to her about her poor parenting and found out whether or not she's willing to step it up? Doesn't sound like she will, though.

Give her a month or two to find a place - you don't have to put her out on the street. Don't let her guilt you into changing your mind. If she loved you so much, she'd care about your comfort and happiness, too, and parent her kids better.

You know what you want to do... you just are being too nice because you don't want to seem like a jerk that kicks a woman and her 3 kids out on the street.

Here's the thing.. it is not your responsibility to be this woman's safety net.  You are not the father of those three boys either.. and whether they have food or a roof over their head is not your problem.

She is afraid of losing her easy gravy train... of course she is going to pull out all the stops to appeal to your emotions.  But, is that fair to YOU???

Look, you are never going to improve the situation doing what you have been doing.  You know her kids are either unfixable.. or she is absolutely unwilling to do the hard parenting work to fix them.  She has made her problems your problems.

But, I am going to let you off the hook here.

She is not the only parent of those boys.  If you break up, and she can't secure a place for the 4 of them???? they can go live with their FATHER ...and she can stay with a friend or other relative or rent a room until she can get on her feet.  It is not your fault that she has "nothing".. that is on HER.  If you have been paying bills.. why hasn't she been saving?  What is she doing with her child support?  I am praying she is still working.. if not.. she needs a job.

The bottom line is that breakups are messy and people try to not be the bad guy.. they want people to like them.. and not be angry at them (mostly women fall in this trap.. but men are not immune).  News flash.. if you are breaking up.. there will be anger.. hurt feelings.. but you HAVE to advocate for yourself.  

I mean... if you think you can manage it.. you might be able to give her 30 days to figure this out... where to go.. but I would not leave my own home.  I would move out of your shared bedroom.. or make her move out.. she can bunk with one of her kids.  But you don't leave your home... in your name.  Be prepared to evict her if necessary.

I know it sounds harsh.. but she is emotionally blackmailing you.. you know you don't want to be with her.. why draw it out and make everyone miserable.  

You need to tell her plainly that you don't see this relationship ever working and the sooner everyone gets used to that the better... and she needs to understand that you are broken up as of right now.. no more hugs..kisses.. no more talking about getting back together.. she needs to put her effort to finding a place for herself to move and move her kids with her or send them to dad's if she can't manage it.  It may not be what she wants.. but she is not the only one with a vote.. and one vote against the relationship can end it.

Nothing is going to change.  Most likely it time to call it quits.  But have one last talk with GF. Telling her to parent her kids.   Where is her ex in this ?  Does he take the kids ?  How much does he takes them 50% of the time? 
You may see a professional for help

I'm a reformed doormat, but even in my most spineless days I couldn't have coped with such a situation.

Your home life sounds incredibly chaotic. I'm guessing it wasn't as noticeable in the intoxicating early days of your relationship, but now you've taken those love goggles off and realized this just isn't sustainable. Sex is probably the only thing you're getting out of this, but the juice just ain't worth the squeeze. 

You know what you need to do, and while breakups are messy, the sooner you do it, the sooner you'll have peace. You can't find the right woman while the wrong one is in your home. As previous posters have suggested, you can give her thirty or sixty days to find a new place, but for goodness sakes stop having sex with her. The last thing you need is her using a pregnancy to trap you.

Guilt and a rescue project are not strong foundational elements for a marriage or a quality long term relationship.

You know this is not your forever person.   Also, it is important to keep in mind that Fragile X syndrome is genetic and a primary cause of developmental disabilities and children being on the autism scale.  All three of her children are developmentally challenged.  There is the distinct possibility that children you might have with her would be at risk for similar issues.

Though it was not originally a motivator for me to not have BKs of my own, I landed on preferring to not have BKs of my own due to an autoimmune disease that I have that would in all likelihood transfer to my biokids.  I am a T-1 diabetic and was Dx'd when I was 16.  I have have lived a very healthy, adventurous, full, and active life and expect to continue to do so for another 2-3 decades (I'm 56).  I have never had an overwhelming drive to spawn though I do love kids.  At least the well behaved progeny of quality parents.  

My DW and I have close friends whose eldest son had major issues that went unDx'd for several years, they had a second son who also had issues.  Eventually they both have been Dx'd with autism.  The jury is out on whether or not they will ever be able to launch.  The eldest who is now in HS might. He was one who was a hellion and very violent when he was a young child.  He and I had a couple of run ins when he took to punching men in the crotch when he was about 8yo.   He got through that phase and is not a bad kid these days.

Sadly, their mom and dad divorced.  Their Dad and I went to university together and my DW and I became close friends with them after they married.  He and I went to work for the same company after graduation and I recruited him to another company.   Their marriage could not tolerate the usual various and sundry stresses plus the challenges of two autistic boys.

You have lived the relationship with this woman and her three autistic kids. Can a relationship where you add joint children to the mix, kids that may have similar issues, be sustained and be a happy one for you?

I recommend that you end it now, claim you home, and get on with life.  The sooner you man up and end it, the sooner you can get on with your life, the sooner your STBX can get on with her life, and the sooner her boys can settle into their new normal.  

Rescue projects rarely if ever work out as quality equity life partnerships.  I have had my own period of rescue project relationships.  The first was one of the most beautiful women I have ever dated. A truly stunning, brilliant, and sweet young woman (I was in my early 20s at that time), who it turned out had a significant drug problem.  She made an effort to reconnect with me when she got out of a long term residential rehab stay.  By then I had moved on...  unfortunately to my adulterous whore of an XW.  After my divorce I dated heavily for a few years which is where I ran into rescue projects #2 and #3.  It turned out that #2 too had a drug problem.  

At that point I transitioned into dating some truly incredible women any one of them could have been a wonderful life partner.  Sprinkled in that phase of my dating life was rescue project #3.  She was a single mom who had just gotten out of a 2yr relationship with her XBF who was not the baby daddy.  She had been out of the relationship with the baby daddy for about 4yrs.  her son was 5-6ish.  That kid was hell on wheels.  She and I had a very intense and passionate connection right away. She is the BFF of the wife of a very close friend of mine who introduced us.  I dated her for several months along with 3-4 other women.  One was the niece of the former governor of the state and a Clinton administration cabinet secretary.  Anyway, rescue project #3 turned out to be a train wreck who actually stood me up for a date the night I called the incredible woman who would be my person/equity life partner.

I had met my DW the week before, I had plans for T-Giving with the former governor's niece's family, who were very old friends of my parents.  I ended up calling my FDW after I was stood up, we picked up some movies, and pretty much spent every minute since making a life together.  I called the governor's niece after that weekend to back out of TG and a couple of weeks later my FDW and I celebrated TG together with my FSS who was 15mos old at that time.  Though on the surface rescue project #3 and my DW had similarities.  Primarily that they were teens when they became single moms, both beautiful though RP#3 was nowhere near the brilliant and amazing woman that my DW is.

I learned from my dating life, failed marriage, and life experience that I will never again over ride my better judgement when it comes to relationships, careers, etc...

So, be good to you, get on with your life.  Set she and her boys free to get on with their lives as you get on with yours.  I am not one to advise avoiding getting to know a person who has prior relationship children.  However, I am one to strongly advise that people should avoid relationships with people with overwhelming past life baggage.  Even if that baggage may not be their fault.  What is theirs, is how they have dealt with their life experiences.  Those who have dealt with it effectively may be true equity life partnership material.  This is one of those you know it when you see it things.

Take care of you. Give your STBX her marching papers so that she can get on with her life and hopefully create a quality life for herself and her special needs boys.

Good luck.

it's very difficult, if not impossible to recapture it.

She's not an effective parent. She is the reason HER KIDS ARE THE WAY THEY ARE.

You are an absolute prize, and dream come true and should be treated as such.

There's a reason she divorced her other husband, and whatever that is, it is not your fault, nor is it your responsibility to look after her and support 4 people when none of them respect you or your home and that is painfully obvious.

You seem like a great guy who tends to try to "rescue" others but who's going to rescue you? YOU ARE.

Please let her know this is not working out and if you need to take legal or law enforcement action to get her and her kids out of your home then please do it. I know this is easier said than it is to put into action, but you must know you will be so much better off without this albatross around your neck.

Should you stay in this relationship, what you are witnessing unfolding before your eyes right now pales in comparison to what you are in for. None of these kids will ever launch, most likely. She will continue to guilt trip you into staying with her, and possibly try to trap you into it by getting pregnant. At that point, it will be exponentially more difficult because you will love your child and not want to give her part-time or full-time custody.

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Don't count on her for birth control. 

However, this point it shouldn't be necessary...time to end this relationship.