After a baby is born, there is a waiting period before you can sign your rights away. In the state I live in, there is a window of time that the birth father came "claim" the baby. I wanted to wait that time, to make sure 'Fred' (or rather FredMom) didn't came back into the picture and mess things up.
Looking back, I bet that is the worst period of time for Family1. I didn't really think about how hard that would be until I had my own children. I suppose they were thinking about "what if she changes her mind", while I had already made up my mind. I wasn't thinking about that, I was concerned about "what if 'Fred' tried to get A1?" Luckily, none of that happened and all was finalized about three months after A1 was born.