Op, i dont if you are still reading this. But, i get tgat you felt ready to move in a month after your marriage ended.
But 7 months is nothing for kids. You had emotionally distanced yourself from your husband years before. Because you had to, as he had done the same.
Your dd is a teen and may not feel the same way. Even if the family wasnt a happy one its still hard for them and 7 months isnt long at all.
I am temptes to agree with dd that you act differently when he is around. Usually around the 6 month mark (when not seeing someone often) we are still showing a our best side. Thats not a criticism, just how it is. But that's quite unnerving for kids.
I think you need to understand its been a difficult year for her. On top of this, she no longer has a relationship with her dad. She is probably scared she will lose both parents.
What concerns me about him is his talk of being a family. Surely the boys are going to be adults by the time you moved intogether and dd is going to be 15/16. They will not view him as a father figure and probably not live as a family for long. It seems like he is feeding you what you want to hear.
You wantes that real family and didnt get it with your husband. This man isnt going to be able to give you that ideal either.
The fact that he has withdrawn is really concerning. His gf, who he claims to want a future with has told him that her 13 year old is unhappy when he is around. This 13 year old has had a negative father figure (op says they were all miserable), now she has no contact with him. Even if he was a shit dad, he is still her dad and thats painful. This 13 year old has recently been through her parents separating.
His response is to sulk and withdraw. He is a grown man. How would this work if you all became a family? There will always be issues if you become a 'family'. And he is going to withdraw everytime she does fall in line with what he wants?
Thats not ok.