Can my husband legitimate our child with my consent if another man signed her birth certificate?

A legitimation is out, because your current husband was never the biological father of your daughter. Your husband can file a stepparent adoption. The biological father of the child must be served with notice of the proceedings. To play it safe, I would also serve the man that signed the birth record as well. If the man that signed the birth record of the child never legitimated the child, the biological father should be served with notice of the adoption. Signing a birth record does not legitimate a child; it merely creates a presumption of paternity that can be overcome with clear and convincing evidence. If neither person can be located for service, service must be done by way of publication. Once the adoption is granted, the legal relationship of parent and child will be created between your daughter and your husband. This appears to be a much better way to deal with your situation. Go see a family law attorney about filing an adoption here.

No. You cannot fix a 16 year old fraud (and a false signing of a birth cerifificate is fraud) this way. You are goiubg to have to convince a judge and the state to change the birth certificate. Georgia does allow petitions to correct birth certificates (you would likely have to serve the man on the birth certificate, and the state requires service as well) although with waiting so many years that may be problematic (I'd need more details to advise on that). And while it may sound strange, a stepparent adoption might be a route. You and your husband need to sit down with a lawyer and share details to figure this one out.

Did you place "the other man" on the birth certificate knowing that he wasn't her father? That's a fraud. Who IS the bio father?

Based on the limited information, it sounds like your current husband wants to become the legal parent of your daughter. If this is correct, then legitimation isn't the right legal tool for you. Rather, you need to consult with a family law attorney regarding the process for him to establish legal parentage, and the implications of doing so.