One bit of advice or perspective I'll offer...
If he is saying he wants the kids only every other weekend, get that in writing, get to a lawyer FAST and get it done.
My AH was a disinterested and still is, father. He is now fighting for 50% custody and will get it bc I am in an insane state with a MANDATORY 50% split custody rule unless there is serious neglect etc... (my word vs his about his drinking and abusiveness isn't proof).
Anyway, I say this to you bc I wish I'd acted swiftly long ago. Once I did take steps to end this marriage and he knew that all I cared about was the girls, he realized that was the one remaining way to hurt me.
Your AH may not want the kids now so I'd seize the opportunity to take him at his word and protect your kids from being with him more frequently.
He's giving you a gift in a way by saying he wants out and wants to be a delinquent father.
My AH would have preferred the life of having a home and kids and wife to come home to but wanted to continue behaving like an a$$. When I said no, he realized that taking the girls 50% of the time from me was the last way to hurt me.
Take your AH at his word and get it legalized and protect yourself in ways I didn't and I will always regret it.
Good for you by the way, for telling people what's going on. When I started to do so for the first time it was empowering and I felt a huge load was lifted from my shoulders.