I was contacted by a family member (cousin) because she is wanting to relinquish custody of her adoptive daughter to my husband and I. We are willing to do so but not sure how to get this done. I know there is paperwork and a court date. We want to fully adopt her so the "previous adoptive family " has no chance to take her back. We are in SC they are in GA. Can I do this without a lawyer? Preferably!

You really can't do it without a lawyer, and the fact that the child is adopted already means even more legal help needed.
Your best bet is to arrange a guardianship for now, which they need to start at Family Court where they live, and they won't need a lawyer for that. Adoption will take a while. Plus the parents need to realize that they won't be able to adopt again, most likely. And they need to experience being away from the child in case they change their minds. You are right that they can't once the adoption is final.

Adoption is a legal process and has to be done through the courts. Because the child was the subject of a previous adoption I would not do this without an attorney.