You are in a tough position there. Between the peverbial rock and a hard place.
Seems more that your stepson is unmotivated, and not planning properly to secure his own home, for himself and his children. That is not your problem, but, I appreciate that you have been lenient and accomodating with him. (Far past the short period of time everybody agreed on him being there.)
I've seen this happen many times where couples end up becoming resentful, stressed out, pressured, and overwhelmed with the responsibility of having to cover financially all the associated costs of having three additional people living at home.
The question you and your husband have to agree upon is, when have you given enough, and what is the next step. That is a perfectly good and reasonable question that should reflect both opinions in one united goal.
Find out what your husband thinks is reasonable. Does he think that he should be allowed to remain another three months, six months, two weeks? Will he agree to allow you to make enquiries on your stepson's situation as far as getting the facts/paperwork/contacts necessary for him to secure housing for himself and his children.
If you can get that far, with both of you agreeing to a time frame, say six weeks, and at the same time, be providing your stepson with everything he needs to move on, and out, then the next part is inevitable.
What if he's still there. What if he has refused to take care of his own business, despite the fact that you and your husband both agreed to a time limit, and finding the information he needs to move out.
Before you and your husband set up a plan to include that possibility, you will find you will both be spinning your wheels, making more adjustments to your life to accomodate your stepson and his children, and you will continue to modify or change your lives, until the next time, and the next.
If your husband is not agreeable to being on the same page, and compromising in a reasonable way to address this huge problem, what can you do. You could go ahead and be the one to have him evicted, but your husband will be resentful most likely, of you.
For now, see if you can't arrange a time with your husband, to go over what needs to be discussed. You may be surprised what his fears are, and how much of that fear is clouding his judgment. Try one more time, just the two of you, to talk, and to listen. Even opening up that avenue to start, is far better than where you are now, which is at opposite sides of the spectrum.
Agree to meet again after that meeting, in two days. Allow him time to think, and realize that you are not being critical of him, but offering a solution to a problem that has to be resolved one way or the other.
You might consider investigating the process where you live, in order to evict your stepson, or research online what that possibility entails. Let your husband know that the two of you together, as a team, must succeed, or, both of you will fail. Be honest and upfront. Expect your husband to be the same.
I get the impression that you've been married a long time. This type of situation is so emotionally challenging (not to mention financially and every other which way), and splits up marriages. That too is also a possibility that you may want to gently tell your husband, is also a possibility, if all other reasonable measures do not work.
Please keep posting. Many have been in the situation you are in now, it is not uncommon for many reasons it seems. But, it is only a temporary problem, or should be, and at some point, your stepson has to put on his big boy pants, and take care of his own business.
But, I also think that without you and your husband systematically taking this problem, together, one step at a time, something is going to change, and drastically. I hope that it is not the end of your marriage.