I wouldn't know much about the US legal system, but it sounds awful, and I don't understand why men get married there - the UK too.
Remember this:
Your kids need you - you are their father and cannot be replaced by some character from a bar or wherever. That pain you are feeling is a manifestation of love, and nobody else can give that to them. They will come to realise this as they get older, especially if their mother is a nasty piece of work.
Also never underestimate the loyalty of kids. Your big fear now is probably that she is trying to brainwash them against you. Kids are not stupid, will eventually see the truth for themselves, and will be furious that they have been mislead.
I have witnessed this kind of thing myself, and it may take time, but karma asserts itself eventually.
Honestly it's a fucking disgrace the way men are treated. Men really need to stick together and fight this shit because it's barbaric. If women were treated the same there would be major outcry. The less you give to lawyers and your ex, the better.