J.B.
Why are you even in this mess? Do yourself and your kids a favor and end this. If you won't do it for yourself, do it for your kids. And don't get into a relationship again until you have the good judgment to choose someone who is a better person. Your husband has clearly failed already as a father if his kids act like they do...are you going to raise your kids with someone like that? And he has no respect for you either, with the carrying on with his ex wife, and by staying in touch with ex-girlfriends before you got married. You should probably work with a counselor to figure out how you missed so many red flags so that you don't get into a mess of a relationship again in the future with someone else.
What on earth is in this for you?
Put together an exit strategy and end this. Tell your kids you're sorry, but this didn't work out and you're sorry to have put them through another transition but you need to put them first and get out of this crazy, dysfunctional relationship.
Normally I suggest marriage counseling but this sounds like it was a mess from day 1 and you went ahead with marrying this guy anyway. You shouldn't have to be checking his phone, looking at his texts, etc. Believe me, been there, done that. Not worth it.