Ex and Cheat are now going out and around my kids

You should move back into the family home ASAP.
Why should you be the one to move out?

Hi OP

There is no legal basis for you objecting to your wife and her new partner spending time with your kids. You know this. Your lawyer does too.

If he were a dangerous criminal like a drug dealer, a person with sex offence convictions etc you could bring it up when arguing custody, assuming you go for shared or full custody. On the basis you have not said he is any of the above, then as a parent you may not like it, but unless you get full custody and deny you wife any visitation rights, you cannot control who she associates with.

and besides he didn't cheat on you. He didn't promise to be faithful till death do us part and sign a marriage contract. Projecting your anger onto him is not healthy. Would a nice guy have an affair with a married woman? Probably not. But then she could have told him the marriage was over, etc etc etc. So of the two of them your estranged wife is guilty of a greater betrayal.

So the only advice i can give you is to trust karma. Cheating is despicable. People who don't have the decency to finish a relationship before embarking on a new one, are not good people. And i believe that the universe, god, karma etc (doesn't matter what your person belief system is) will see & judge peoples actions and that in time they will get their just rewards. It is this belief, that allows us to not feel the need to personally punish people.

Your only concern now is to ensure you children have a stable happy home life. do what you can to promote this. Being civil with your wife and her partner may seem a big step, but i'm sure that you can see it will benefit your children. And when your head is clear this is the course of action you should take.