Yeah, he sure sounds like the very very typical person in an affair. If he works at a school it should be fairly easy to narrow down who the candidates are? (Unless the students are adults, then all bets are off). Has he even told you the name of this "friend" he is supposedly moving in with? (And please please tell me this school isn't a high school and we don't have to worry about him sleeping with a student?)
And on the vacation - heck no. YOU deserve to go and have a good time with your kids, he gave up rights to that when he decided to have an affair and move out. Bring a good friend or sibling with you and make it a great trip. Let him miss you and the kids.
Really, as hard as it is to watch them drive the bus off the cliff, you cannot stop him from his craziness at this point. You can tell him what he's giving up, you have offered to go to counseling with him, but in the long run, he's going to do what he's going to do and there's very little you can do to affect that except be the person he would be crazy to leave. Create a fantastic life for you and your kids.
Part of my story is, my ex cheated the night before our wedding. I forgave him and put it down to wedding jitters. When our 3 children were young teens he cheated again. I DB'd my behind off and we reconciled, and actually had several very good years. Then when my youngest was 17 he departed in a MLC cloud as he was turning 50.
In retrospect, my children suffered a lot because of my poor choice to make babies with someone who had already shown he could cheat. And in those last years between his affair and his departure, I thought things were good for my kids - but they told me later they were always waiting for the other shoe to drop. It took me some time after we separated to realize what a narcissist my ex was and how much of myself I had given up in trying to keep him happy and accommodate him. I blew past SO MANY red flags in our relationship.
I'm not saying he might not come back - he very well might, this might be a temporary derangement. Once he finds out OW can't make him happy either because HE is the problem, he might wake up. BUT - the best way for that to happen is for you to move forward with your life, dropping the rope. Make life for you and your kids so fantastic that he'll be begging to come back - and then make him do the work. (IF you even want him by that point, and you very well may not).
Do you have family in the area? What kind of support system do you have? Where is his family?