You absolutely have choices. The question is do you want to stay with this lying cheating ass?
As the others have said, see a lawyer. File for divorce. Let your lawyer do the work. Have this woman's name in case it's needed. As someone here already stated though...family court doesn't care if a parent is a cheater. Keep any documention he sends you, and you should ONLY send communication regarding the children and finances.
I know you're worried about custody. It sounds like he's the type of man who would fight for the kids, just to hurt you. So let the courts handle this. Don't let him see that it's getting to you. If he wants to see the kids, ABSOLUTELY. Mention how you have plans, and he needs to be on time. Call him (if you're separated) and tell him you need HIM to take the kids because you have things to do. Turn this around on him.
He probably wants his life free to be the fun single guy...wake his ass up to what it will be like to be a single father with responsibilities that take him away from the OW at work. He will find out soon enough, it's not all fun and games.
I'm sorry you're going through this. Hang in there and realize you have far more control than you think...