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.
You should get family law counsel to draft and get completed an affidavit of relinquishment of parental rights by the mother. Your attorney then will file a petition for termination and adoption to get her rights terminated, file motions and orders for riminal background checks and home study Your attorney needs to schedule a trial date. Amended birth record paperwork is done. An attorney under the facts described should be able to do your case in range of $2,500 to $3,500. Court costs are filing fee, social study, criminal background check and Vital Statistics birth record amendment. A termination-adoption needs to be done correctly for the child's sake.
Now if you wish to go it alone, there are ways to do so. One such way is to utilize the services of a Do-it-yourself adoption service such as RapidAdoption.com. For a reasonable fee, the fine folks at RapidAdoption.com will complete all of the necessary paperwork for the adoption in your state. They will even do it if the child is in a different state than you. They have been doing legal paperwork since 1998 and have prepared 1000’s of legal documents in all 50 states. But their service is only available in the United States and only with domestic adoptions. So if you reside outside of the United States or are adopting from another country, then this is not an option for you.