I just had to reply. I can totally understand how you feel!
Let me preface this by saying I'm not religious. Not to say I dont believe in a God of some sort, but I'm definately not Christian.
In any case, I really felt like I was reading my own story! Well, of course with some variation but still! My ex also has a lot of mental health problems, but he has seen many psychologists/psychiatrists for it and is on meds. We also have 3 children together (my eldest isnt biologically his; he was an oops, but he was a father to him from the time he was 18 months old).
Much like you, many of the bills used to be in my name. He didnt pay them, stuff got shut off, and then he used his name to get the things he wanted back. He worked off and on, we'd go months with no income, which was rather scary. He also went years without working, so I'm not going to lie, that was an improvement. However, I found that even once he had a good paying job, bills still didnt get paid, I rarely saw any of the money, and all the promises I was repeatedly given never happened. He didnt help with the house or the kids and like you, I couldnt leave them longer than an hour, if that. Now, I understood when he was working, I had no problem taking care of the house when that was going on. But when he'd be home and not lifting a finger-even after I had HELLP syndrome and had a cesarean-that to me, was unacceptable.
We've broken up and gotten back together many times. We were trying to work through things when he harassed me to no end until I finally caved and agreed to another child before I was ready (the one that I had a cesarean with). To me, anyone who truely loves their wife and really wants to work on a relationship, would NEVER push someone into having a CHILD. I mean...this isnt a kitten, its a PERSON. In any case, it kind of spoke to me as time went on, especially as I started getting blamed for more and more of our issues. It went so far that I was told the reason I couldnt spend any money was because "you didnt spend it fast enough...." I mean really. And of course, there was the fact that he never got a minivan or anything like that, meaning I couldnt go anywhere-there wasnt enough seats in the car.
Anyway! The point is, I tried and tried to work stuff out. I tried to put all that past me. And had he actually really attempted to change, I think it was possible that I could love him again (by the time I had my last child, any love that was there was completely gone) or, at least, lived with him and kept my family together. But, well, he didnt. I gave him YEARS to change, over and over, and nothing happened. I finally said enough and got divorced a few months ago. Now I'm with a man that I love more than anything and does everything for me-or with me. A man that does dishes!

I'm not trying to say this will happen with you. This is just the begining for you. Let him try to change, I really hope he does! Remember that it can take some time, but at the same time, it cannot take forever. I dont know what your threshold will be-but I went through homelessness, having no money, no car, trying to go to school and work to support him while he didnt work...I like to think my threshold was pretty high! I know its hard, I know that people CAN change, but also know that you can make it on your own as well.
Oh also, I agree that when you're in your current situation, you feel pretty helpless, but it WILL get better. Once you have income, get a car, etc, well...everything changes. Like I said, its only been 3 months for me and I'm happy as can be. Still worried about money, but I'm so happy I finally got out.
Good luck and keep us updated on what happens! *hugs*