There are two stages to divorce -- the legal divorce and the emotional divorce. it sounds to me as if there's some trouble with the latter. . . .
and, honestly, I don't think you are going to be able to get past this without some kind of argument. Maybe over the phone? (to make it easier on you). When he suggests you get together, tell him that this is very difficult for you, and that it's not fair to the children either, because it helps to keep the dream alive that Mommy and Daddy might get back together. It's really healthier to make the break and make it stick.
That said, it's hard. But it's better to have the dream die for mom and dad, as well as for the kids.
My ex had a girlfriend within 2 weeks of when we separated, yet he kept wanting to get back together for about 3 or 4 years -- until I found someone serious to date and then marry. And every so often he would be "so wonderful" that I kept wavering and thinking that maybe we could actually get back together and make it work. It was hard, because although I divorced him, I liked the guy. He was fun. He just wasn't very responsible, didn't make the family his major focus, etc.
Lesson learned: break it off, but keep the contacts as positive as you can, and try not to complain about him or highlight his bad points in front of the kids. Give them every opportunity to love their father without feeling disloyal doing so. Let them have positive encounters with dad, and keep your own relationship friendly enough (if you can) so you will be able to discuss problems the kids have as they grow up.
It's never an easy road, and it's hard to put down all the dreams as divorce becomes final. and you're going to need to draw a boundary line for both of you since he's working hard on not having one, even if he does have a girlfriend.
good luck. Not easy. But draw the line and face the struggle. The longer you put it off, the more embedded the non-family family days will become and it'll be harder on all of you, esp the kids, when the habit dies.