When my husband adopts my son, does his biological dad have to sign over his rights even if he's not on my son's birth certificate?

You need to speak to a family law attorney. I do not believe that your husband can adopt your son while his biological father still has parental rights; your son's biological father's parental rights need to be terminated, whether voluntarily or by court order, before he can be adopted. (Think of it this way: even if emotionally you can have two fathers, legally, you can only have one at any given time.) You need to advice and assistance of a family law attorney for this situation. Good luck.