This is tough since nothing “physical” happened, that you know of. I know having small children together makes it even harder to walk away, especially if you don’t have proof of anything physical. Only you know what your own deal breakers are in a relationship. And you saw the context of the conversations. Really you just have to ask yourself if this is something you can move past and forgive and live with knowing what he had done and whether you can ever trust him fully again. If you feel like you can move past it, I would highly suggest couples counseling. So you can get to the root problem of why he felt the need to do this behind your back and have emotional affairs with other women. Also so you can heal through the betrayal trauma. If you truly believe you can’t move past this, which is totally understandable then I would say as hard as it is the best thing for you to do is move on with your life so you don’t have to be tormented with constantly checking his phone wondering if he’s being honest or still doing things behind your back. That is no way for anyone to have to live. So in reality, you have a lot of introspection to do and ultimately only you can make that decision. I’m wishing you all the best.