From your post, it sounds like you want to do a private adoption without the birthparents involvement. Unfortunately, it's not possible to go around the birthparents' rights. They have to be notified of your intention to adopt & you have to have their consent. And like I said, if you want to go the "I can't find them" route, it has to be an honest "I hired a private investigator and here is everything we've done" kind of situation. You have to convince a judge (who doesn't terminate parental rights on a whim) that you did everything humanly possible to contact that birthmom
When my wife and I adopted our son, we were told that we needed to hire multiple attorneys. One to represent the birth mother. One to represent the child. One to represent the birth father. And one to represent us. When we adopted our daughter we needed even more. One for the child. One for the birth mother. One for us. And one for each of the three possible birth fathers! We were outnumbered! Since no agency was involved, the attorney fees, the home study and the court costs were our only expenses. And when it came time to file our income taxes we claimed the adoption tax credit and were reimbursed the entire amount. Both times.  We were fortunate. Adoptions can be, and in most cases are much more expensive.