Should I Reconsider

Me and my boyfriend have been together for a few months. And I got pregnant before we got together. And as time has gone on he has asked me to adopt my son an sign his birth certificate an be his father as seeing his bio dad want be in his life wants nothing to do with him. But when we are around people now he says “he isn’t biological mine but I will be taking care of him like he is mine”. Like he has to make sure people know that this isn’t really his baby. An I just keep thinking to my self should I reconsider letting him sign my son birth certificate or keep with the plan because he is going to here for him an take care of him like that’s his son. We live together an he is really excited about him coming an he has been here since we started dating. Doctors appointment, ultrasound and everything that just comes with being pregnant.

Now I’m thinking like should I not do this an just see how it goes An do it in time. Because he could still walk away at any point.