You do not ever have to answer the question "is it his dual diagnosis, or is it his alcoholism?" You don't ever have to sort out "why" he behaves the way he does.
The real questions, as some articulated above, is "how to you want to live, and what environment do you want your children to grow up in?"
Right now, you have placed the ball in his court, and are waiting on his timetable for his behavior to change so that you feel comfortable with the life he is creating for you and your children. He has all the power; he controls the timeline to get to health or not, while you wait. Your life is frozen until he works out his issues and changes. You are stuck with whatever he does. And he may not want to or be able to change in a way that is acceptable to you.
I know if it was me struggling I wouldn't want him to quit on me (and by now he would have), but at the same time I feel like I am at a crossroads.
I suggest re-framing this idea. In my view, the children in the family deserve top billing. This isn't about your quitting on him; it is about you needing to be the single parent who can create a healthy environment for your children to grow up in. He is accountable for being a good parent as well as a good partner.
From my experience growing up in a severely dysfunctional environment, children easily believe that they are at fault when the problem is that the parents' behavior is flawed. The childrens' emotional health needs to be at the top of the list.
Try turning your thinking around the other way. Think about what you need to have a healthy home life for you and your children. Articulate the different elements; describe to yourself how a happy home life will feel and will be.
Then, tell your husband what you need. If it were me, I would ask him to leave until he can provide the security, serenity, and safety (emotional as well as physical) that you require for yourself and your children. Your children deserve two good parents, or at the least, one good parent and the absence of a disruptive damaging parent.
If your boundary is "I will not live with a man who is "XYZ", then, you can take appropriate action and your life and your childrens' lives can get healthier starting today. The ball that is in his court is whether he wants to - or can - change. You are not trapped in a bad situation while he has the freedom and the to either change or not change, and bear the consequences of his actions. And when - if - his behavior is constructive, he can come back.
He can work hard to get well; he can spend time understanding what it is to be a healthy parent; he can go through the medication trials and tinkering on his own without damaging your children. Or, if he doesn't, he bears the consequences, not you and your kids.
Take what you want of this, and leave the rest. Good luck to you, and I hope you come here as often as you need; you'll find a great support community here and many who have gone through similar difficulties.
ShootingStar1