No bu your SO absolutely MUST prove to you that he can be a real parent to his son by ensuring he completes his chores,keeps his room and bathroom spotless, and that he is respectful and polite. He also needs to lay off of your daut, he is not her father. He needs to do all of the above for at least a year, if he can't do that well ... his loss.

Draw your line in the sand and make your limits heard 

He says he doesn't want to lose you, but is he showing you that? The only way things will get better is if HE makes changes. Is he willing to do that? Perhaps you need to lay your cards on the table:

X, Y, Z must change or I am leaving. Hard line and you will find out real quick if he is going to make those changes.

He is a 17 yo who should be leaving in a year or so correct?  it sounds like you married a guy that was not a great role model for his son.  they both lived in squalor. if your SO loves you. can he learn to be neater? maybe.  can he learn to insist his son is neater? yes. but will he? that is less certain

It does take time to change. But what is he doing to catalyze the change?

Is he engaged in therapy with or without you? Or with his son, perhaps? Is he taking parenting classes, or otherwise seeking out a parenting coach? Even reading books about parenting teens? Is he having his son evaluated for any mental health issues?

Is he setting expectations for his son for an independent life? Is the son prepared to live on his own and keep a job of any type? If not, what is your SO doing to prepare him? From what I see, you have a lazy SO and a lazy SS and neither are motivated to change.

If your SO does nothing, or easily gives up, then, no, nothing will change for you. His kids will always be dependent on him. Maybe your SO is keeping his manchild depdent on him for a reason, filling some psychological need? So we're back to therapy for him. WIll he go and do the work?

"[He] has expressed that he doesn't ever want to lose me."

Unfortunately, words are cheap.  What has he done to SHOW you that he doesn't want to lose you?  You've told him what you need.  What has he done to effect that? If he's not punishing his son and making sure all the chores are done, then he's less concerned about losing you than he says. Perhaps what he really means is "I never want to lose you as long as I don't have to do anything difficult or inconvenient to keep you."

Is he financially in a position to get a small apartment where he and his son can live in squalor until the son launches?  Of course, the children of failed parents don't always launch, but that would at least give you a clean, peaceful place to live while you see if SO can change and/or the son can launch. 

Ask him about his childhood and how he related to his parents.  See if you find any patterns to currrnt life.  

SS is 17, right? HE should leave within 1-3 years, IMO. If he's going to college, maybe there is some financial aid that could help with living expenses. If no college or other training, full-time work or military after high school. 

I feel like men need to have things broken down into the simplest terms possible. When my SO and I moved into our home with his bio kids it was a big adjustment. And we are no where near a sunshine and rainbow ending.

BUT I was begging him to help me in the household. I said I needed more and didn't want to feel like the maid picking up after everyone. Had that same argument for months. It was insanity. Until I finally said I need this list of things done, WITHOUT me asking! Since then he has changed big time with the household stuff. I don't ever have to ask anymore thank goodness.

But we still have a ton of issues when it comes to parenting and boundaries. I have no kids so it's hard for me. And we have very different parenting styles. I suggested therapy and he REFUSED and REFUSED my boundaries I asked for. It wasn't until I told him I was going to look for someplace else to live that he agreed to therapy.

I am hopeful we can learn of tools to help us. But you have to be willing to stand for what you want and need! My therapist said to me one day (I go to personal sessions on my own) "you're already miserable, what do you have to lose?" And she's right. I'm already the bad guy in my house and feel horrible about my situation. It can't get much worse. I'm wishing you the very best! Stay strong, you got this! But the effort needs to come from him! 

There is another option; for you to accept the situation as it is. Unfortunately most of us are unable to do that, especially because it conflicts with how we parent our own kids. Your biggest issue is your fiancé. He created the monster that is his son. It was them 2, living that way, and SS with barely any expectations, and now you enter the picture and want it to change, but they may want it to stay the same (because maybe they are codependent). 
I can only share my journey with you, I went through it twice with my DHs kids because he had custody of his first 2 sons. 
ss23 was worse than your ss, he used to take 3 hour baths, and kept a filthy room. They both used to stuff garbage between the sofa cushions. At least he graduated, but then even after he found full time work, he struggled to help us in our household (100 weekly). He always used BM as an excuse for why he couldn't save, but really it was his marijuana habit and partying. DH was a doorman so on his overnight shifts, Ss23 would bring girls over and his friends, who were gang affiliated. Last year he was asked to leave because it was 4 years of that crap and I had to put my foot down to DH. I had him force save his entire check so he can move out and after 3 weeks of that, SS asked for his money back. dH told him, "you understand that you have to leave if I give you your money back?" And ss said yes. By this time he had met his current gf who has her own place by way of her father so I'm sure he gave her a cockamamie story about getting kicked out so he could go live with her. 
This is why I don't have the patience for Ss18. I'm not going to go through another 4 years of him possibly getting a job, possibly not saving. He already smokes marijuana, and stays in his room all day. While he doesn't have a gf or many friends, it is still annoying to have to come home from work and see that the bare minimum has been done. I don't want him to live with me, and if that came at the cost of my marriage I was ready to pay it, because I had a little taste of what it would be like just Dd8 and I, and it was peaceful. 
We can only speak from our journey, but I wish you lots of luck. The good news is that there shouldn't be much longer of him living with you, but that's depending on if that's what your DH truly wants. Have you asked him what he wants SS to do after high school? 

If you can't make him clean up after himself. You are going to make him get a J O B. And actually work. ??.  DONT THINK SO,   Unless you light a fire under SO and make him a man. You will be raising three kids

There's a lot of intelligence, experience on the site giving you advice. 

Some advice is biased, some isn't.

Some advice is projection, some isn't.

I could go on.

But when the large majority state the same thing, and there's basically consensus, even among the most unbiased, you should listen closely and reflect strongly on that advice.

Your partner needs to take action. Like others above, give him what needs to change and a time limit to show you he's ready to step up and do what needs to be done. While he's showing you, prepare a plan of exit so if/when he doesn't change by your deadline, you are prepared. Always stay ahead.

less.

Demand the change, put it all on him, and never tolerate anything less than him doing exactly what you tell him to do when you tell him to do it. Including his immediate control of his failed family toxic progeny.

Period. Dot!

Failed parents do not just stop being failed parents when their failed parenting experiments turn 18.  This is in all liklihood a life sentense for you and for your own children.

He has proven himself to be a failed parent. Minimize the exposure of your own children, even the one you share genetially, to this failed parent and his toxic failed family progeny. 

His continuedpresence in your life, relationship, and his presence in the lives of your children should only be as long as he immediately is cured of his parental Cranio-Rectitis and has zero relapses. One relapse, and he has to be gone and stay gone.

IMHO.

Your optimism is a high risk to yourself and your childen.

Be very cautious.

Good luck.