Even if he wasn't cheating, he sounds exceptionally immature. He blamed you for his cheating? Sheesh. He's a big boy. That means he needs to take responsibility for whatever he does.
I assume you had a divorce lawyer? If so, see if you can pay for an hour or two to get some help with drafting a custody agreement and a trip to court to get it signed by a judge. Get all of the custody and visitation stuff in writing and signed by the court so that it has to be followed. If you follow it, then no matter what he bitches about, you're following the order. And if he wants it changed, then he gets to go to court and see if it can be changed. But that takes it out of the equation and you won't have to "figure it out" because it will have already been figured out.
You've got a new place. Stay there, at least for now.
This guy sounds like he's used to getting his way, and getting it quickly. So give it some time living apart. As in six months to a year. And then see what happens. Has he moved on? Have you? Has he changed his ways?
You're divorced. You can date and talk to whoever you want to, and so can he. But if he is giving you a line about wanting to make things work out, then he has got to show you rather than tell you that.
His actions will speak louder than his words, as they should.