Why are you hanging around?
You have proof of … something. Certainly you have proof of lying, and potentially proof of a coverup. But you already don't trust her. And she's already walking on eggshells around you.
What do you need? Do you need to find hotel room receipts? His hair in your bed? An empty box of condoms?
Playing junior detective isn't doing anyone any good here. If you want to stay married, then find a way to forgive her or at least stop hounding her. And recognize that this will probably continue unless you two get into couples counseling and you both
go into it with 100% conviction. Go all in or go home.
If you don't want to stay married, then start making moves in that area. But also recognize that maybe you're overreacting—although I honestly doubt that you are. And recognize that ending a marriage is an expensive proposition. But it's not impossible.
This in-between stuff is for the birds. Staying together in this sort of limbo is unhealthy. And if you claim you're staying together for children, then I got news for you. Apart from the tiniest of infants or children who are extremely mentally challenged, your kids know something is going on. They may not know the details, and they may even blame themselves. But they understand things aren't smooth or perfect. Don't kid yourself and pretend you can conceal it from them and somehow protect them. You can't.
And the jig is up, anyway. They know something's off.
If couples counseling isn't in the cards, then seek it out for yourself. Why? To figure out how to handle your anger and hurt, and make a plan for what'll happen next. Whether you step into the next decades of your lives together or apart, it will help to have a plan.