Actually from what I gather it is quite easy [sometimes] to adopt and a birthparent never even know. This is particularly true with birth fathers. In our adoption case (with my now adopted son) the absent birth parent was willing to sign and so it was quite easy. I have heard of many cases though in my research down the adoption path where the adoptive parents did searches with the OAG, DMV (as mentioned), used paid internet investigator searches, and even hired PI's then presented the evidence to the court and had the TPR signed. The truth is that none of those routes yield astounding results necessarily. I don't agree that the birth parents (the fathers in particular because these are the ones that it usually happens to) are being treated fairly but it does happen. Hmm, I guess they aren't really being treated badly they are just not being found. Maybe they don't even know? And, of course the cases I'm aware of are dealing primarily with step-adoption because this is what I'm familiar with being that it was my situation. Judges are far more leniant in these cases I imagine because at least one biological parent is present. I don't know for sure but it would be my guess.
You need to have a lawyer. You can't involuntarily reliquish. Relinquish means your are voluntarily giving up the child. Both parents have to voluntarily relinquish or have involuntary TPR done by a judge in court. The birthparents rights have to be terminated by a judge before you can begin the process to adopt. Usually an adoption agency handles the TPR signing. With an independent adoption, you have to have an attorney. First they will locate the birthmom & have her sign consent. Then they will attempt to locate the birthfather. If they don't know the whereabouts of the birthparents, they will have to make some kind of attempt to locate them. Usually it means sending a certified letter to the last known address, they can contact motor vehicles for address on a driver's license, and/or put a notice in newspapers in the areas where the birthparents might be living. If no one comes forward, the judge will usually TPR the parents based on abandonment and then you can adopt. The judge will need a homestudy since you are adopting a child who is not a relative of yours.
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.