I second everything that BearlyBreathing said.
He's doing something called blame-shifting because he knows that what he did was wrong, but he doesn't want to accept responsiblity for it. His decision to hide the truth from you is entirely on him. There are other ways he could've handled it if he anticipated that you wouldn't like what he did, but deep down, I think he knew that he was doing something wrong. That's usually why people hide things and/or lie about things.
In your signature, you say that dday was a year ago. Have you spoken to any lawyers since then to understand the full picture of your situation? There are alternatives to divorce in many places. You can live separately and remain married if you and your children are dependent on his benefits and income. In house separation is hard, and I won't say that it gets easier with time, but it's doable as a temporary solution.
You know best about the details of your situation. I stayed for a couple years because my ND kid was having a rough time and I didn't think destabilizing her home life would be good for her. By the time I filed for divorce, she was doing a lot better, and she's handling it all very well (age 16). I was really scared of what the news would do to her, but she wasn't surprised about by it or particularly upset about it. STBWX and I didn't fight or anything (she overheard nothing), but she'd noticed our growing distance and little things, like us not wearing our wedding rings. Don't underestimate your kids.
And finally, I'll say that I wasn't able to be as resilient or present for my kid in those 2 years of R as I was before dday or as I am now. It's hard when you're miserable, grieving silently, and especially when you have an unremorseful partner. You don't have as much emotional bandwidth, and you have way less energy for your kids or work or other aspects of your life.
You sound like a loving, thoughtful parent who's willing to sacrifice for her kids, and that's great. Think carefully about whether the current situation is really what's best for them, and if it is, do your best to detach from your partner. Don't try to change him. Don't try to convince him about his EA or anything. Don't monitor his travel plans or behavior. You have to stop loving him the way you used to. That is very hard, but it's doable with practice. Practice accepting that the marriage you thought you had is over. Practice accepting that he is not the man you thought he was. Practice accepting that he is a deceiver and not to be trusted. Avoid sex and other forms of intimacy with him. In your heart, start thinking of yourself as divorced. Remind yourself that today is not forever, and that eventually, you will be able to change your circumstances.
Things can get easier, but nothing changes until something changes. When you're ready to make those changes and take control of your life, you'll start to feel better. Hang in there, 4bearsmum!
[This message edited by NoThanksForTheMemories at 11:14 PM, Friday, May 29th]