Bring this to your pastor, who I take it you are working with like a counselor. If, however, you would consider a couples therapist, that could be another option.
But either way, use sunshine as a disinfectant. Right now, it's been secretive and that may be a part of why your husband is getting this kind of a high from things. So get it out in the open with your pastor and hash it out as well as you can.
And make your feelings known. I am assuming you are hurt (I know I would be). And maybe angry. Or feeling betrayed, etc. I don't want to tell you your feelings, of course. Those are for you to feel. But get them out there. Don't just take it.
Your husband needs to hear it, not just from you, but also from your pastor, that his behavior is unacceptable. He said he would change and would cut off all contact, and he has not. And that is a corollary betrayal on top of the initial affair betrayal.
I would also suggest you talk about money when the three of you meet. Find out just how much your husband has sunk into this affair. Because flying someone across the country ain't cheap.
In the meantime, I would also suggest talking to your pastor on your own. And see what he (she?) says. Because I know you believe in your vows and I do respect that, but you are placing yourself into a tough position if certain responses are utterly off the table. Maybe they don't have to be, or you can do something else. Plenty of people co-parent effectively in a separation situation.
And of course your children come first – but you are not doing them any favors by ignoring the elephant in the room. They are absorbing lessons about interpersonal relationships from both of you. It's up to the both of you whether you want those lessons to be about taking a lot of guff for the ultimate purpose of staying together, or about being independent, or about standing up for yourself, or something else. You have the power to direct this narrative.
Best of luck to you.