He can't sign away his rights unless the mother agrees to it. If she agrees, then it will be a voluntary relinquishment of parental rights.
It is not always necessary for there to be a stepparent adopt in NC. This also does not go before a judge, it goes through the Clerk of Court.
As an attorney told me, it is not about making children fatherless, they don't care about that. It is often assumed that it is about making the children fatherless but it is really about the money. The state and court are scared to death that if someone's rights are terminated that the other parent won't be able to support them and the child will end up on state aid. I already knew that before the attorney told me because I think that's pretty obvious. It's always about the money. That's why they are scared to terminate the rights unless there is an adoption because at least then there will be someone to go after for the money if they should split up.
Right now your hands are tied unless the mother agrees to this. If she does, she will have to be the one to file for it. Yes, it can be done without an attorney. The papers to file can be picked up at the courthouse but if you can get an attorney, do it!