Good thoughts from everybody, but this is going to be hard.
Just a few things I learned - I didn't have to deal with the issue of my spouse being in a relationship already - that complicates it a lot.
When we separated my kids were S8 and D10 (and two adult stepkids). It was hard on us all. I got the kids into counseling shortly after we separated and that helped some.
If you are still sharing a house in any way, I would suggest you try to end that as soon as possible. Let him know that everything will go better - and you'll try to help him have his business - if he will live somewhere else. Being in the house together while neither of you is committed to the relationship, and you know he's seeing someone else, will just be way too stressful for everyone.
Minimize communication with him - it will only make things worse. Don't try to be friends. Use e-mail because it leaves a record. He has made his choice and hasn't treated you with respect, so he's not someone you can trust. Deal with him on a business basis - no emotions, just facts, like schedules and other information. If he can't deal with that, let it be his problem - don't engage him on any basis other than business.
Find an attorney with experience in cases like yours - "high-conflict" divorces. Ask for "war stories" showing that she has that kind of experience and has learned from it.
Think about what you believe would be the best realistic outcome. Learn about how your state does things; for example, in some states, it's almost always "shared custody", but one parent might have the kids most of the time, and the other maybe just "EOW" - every other weekend. Your attorney should know what is realistic. Develop a proposal with reasons based on what's best for the kids.
Tell your kids the truth, but not too much. Best to talk with each of them alone, and just share the basics, and see what she asks. My kids always asked "basic" questions - not philosophy, but basic stuff like, "Will we still take the same bus to school?" - how events affect them.
Finally, I would suggest you read "Splitting" by William A. Eddy, about divorcing a "high-conflict" person. You can also find good articles on his web site,
www.HighConflictInstitute.com.