He said he was coming to stay the night tonight. I think I need to leave while he's here. I don't have any plans and most of my friends are busy so I'm not sure what I'll do.
Go out to a movie? Get a little dressed up and treat yourself to a nice cocktail or glass of wine or Shirley Temple?
I might lean toward letting him see that you are out and about and coming home late as opposed to staying elsewhere for the night. To me, the first is more genuine and the latter might look like game playing/pursuit.
Plus you'd be taking advantage of the one singular perk of him making this decision - he'll be 100% responsible for the kids and that means you get a break and get to start rebuilding a sometimes-kid-free social life.
Are you thinking of a longer term plan, T0? It's good that he's coming over to take care of the boys (because he can see your progress and the boys have stability) but at some point allowing him to come stay and do his parenting time when he prefers and in your house is shielding him from consequences.
At some point, there should be a shift to a set schedule and him taking the kiddos. (I feel more strongly about the set schedule - so you don't end up in a situation where he's all "Hmm, I'll go to this concert, and then three Tuesdays from now I don't have plans so I'll watch the kids" because that is cake eating. He doesn't get to pick and choose based on what is most convenient for his new social life. You need predictability and so do the kids.)
I strongly suggest that once things have settled, you be the person who says "H, now that you're moved out and established, the boys will do best with a set schedule. Plus it allows both of us to plan. Here's what I'm thinking will work best. Review it and let me know what you think." This action is very clearly Not Pursuing, because you are accepting his decisions and actively creating more distance.
He wanted to be a part-time single dad, so let him. That comes with some restrictions and less freedom than he enjoyed while married to you, since you served as the primary caretaker. Let him get a taste of the responsibility!