How long have you been married?
Has he ever cheated before that you know of?
How old are the two of you?
When is the last time the two of you were intimate?
I find that there's no cheating "type". Not with men or women, because it's always the ones you think won't cheat, that wind up doing it. I'm sorry for what you're going through, but I must say that his moving out is a very good indication that he's likely keeping up the affair with this other woman, additionally its not indicative of wanting to work things out. He's basically showing you (and the children) that he's made a complete break. I'm also concerned that you said he cringes from you, thats not good at all.
If he says he's going to file for divorce, then think about taking it seriously and think about how you will move on from this. If he's willing to go to counseling then there may be a chance of working things out, but it seems like he's not putting in as much effort as he should, which is certainly not fair to you, but without counseling he may not know or care about that.
I welcome your thoughts, let me know if you want to talk more.
If it will help you to get closure and move on, then I would say yes. If this person is telling you that they may want to work things out, and has you hanging on a string while they do whatever they want, then it is completely up to you if you want to find out the truth. As they say the truth will set you free...maybe this will be something that can help you move forward and end his lies once and for all. Of course if you want to talk more I am here.