I would be VERY careful and hire an attorney
Let me say first that I am not an attorney in GA. I have been an attorney elsewhere and I have TONS of experience both personal and professional in this area. I was doing research on another topic when I stumbled upon this forum. When I saw this post I just had to register to answer this question.
First, I would be extremely skeptical of taking advice from unknown people on this board, even me. GET AN ATTORNEY!!!! Do NOT try to do this the cheap way or you will end up with no rights AND child support. Believe me it is possible. This is not going to be cheap. I am sure you don't want to hear this but if that child is his a LOT of his money is going to go to dealing with the child. A custody and child support case alone if he doesn't fight it will run $1,500-2,500. If he has to fight for his rights $5,000 is pretty much a MINIMUM. Then, if the child is his you can be certain he will be paying monthly for the next 18 years.
First lets clear up somethings that are just wrong. One poster said if you didn't sign the birth certificate you have no rights to give up. That is painfully wrong. As the biological father you have a right to legitimate the child, have visitation, and become a part of that child's life. That right can be ignored long enough that it goes away. If you want to know more about that see IN RE BABY GIRL EASON., 257 Ga. 292, 358 S.E.2.d 459 (1987). I would post a link but apparently you have to have 10 posts here to do that???? Google it you will find a copy of the ruling. In any case you have rights to give up.
MUCH more importantly you have obligations that can be enforced against you. I don't know why you say DHR is suing your fiancée for $450.00 a month since the child's birth. That generally isn't how it works. If DHR sues you, or if they have, you have to respond within a certain amount of time with financial records and they will determine how much child support you will owe by federal guidelines. YOU CAN NOT TERMINATE YOUR RIGHTS TO GET OUT OF THIS CHILD SUPPORT. If you don't respond in a timely manner you will be SCREWED. Do not fail to respond.
The only reason DHR would be involved without a current delinquent support order is because the Mother is receiving TANF (Temporary Assistance for Needy Families) and probably Medicaid. DHR can come after the costs of the birth and they will attempt to recoup assistance they have given to her including the cost of the birth. If they get a judgment and he refuses to pay his driver's license, professional licenses, tax refunds, and liberty will be forfeited.
From that point forward you will also have to pay child support unless and until someone adopts the child, the child is emancipated, or the child turns 18 and does not go to college. Your fiancee's lack of concern for this child isn't admirable. He will not be able to skip out on that responsibility by just signing away the child.
Duranie is closer to being right but is still wrong. You CAN give up your rights to a child, but you CANNOT give up your obligations to a child even with the Mother's permission. If you are not careful, you will end up with NO RIGHTS and ONLY OBLIGATIONS.
Let me repeat: GET A LAWYER FAST and don't try to avoid the cost by winging it.
The others are right about Paternity testing. First he needs to know the child is his. DNA ! If DHR is involved and you dispute paternity they will have him tested. If he is the Father he will be billed for the test (about $250). If the child is not his he is "off the hook."
If the child is his, he is OBLIGATED to support the child and to reimburse DHR for some of the benefits she has collected for the child. She will also be OBLIGATED to give him access to the child unless he signs away his rights and keeps his obligations.
Your fiancée should want to be in this child's life if he/she is really his. He obviously doesn't want to be in the child's life or he would have pursued his legal rights sooner. Maybe he didn't know how, and I'm being harsh. But if you are going to marry this man, and this baby is his, you are going to have to get use to the idea that he won't be all yours. That baby MUST be taken care of.