Taking a scientific paternity test does not in and of itself give you rights to the child nor does signing a birth certificate. You must file an action with the Court to request your rights and responsibilities be outlined in a judgment.
The natural guardians statute provides that in the case of a child born out of wedlock the mother is the natural guardian and is entitled to primary residential care and custody of the child unless a court enters an order stating otherwise. If the mother objects to your having or seeing the child she can make doing so very difficult (absent a court order).
IF YOU LIKE THIS ANSWER AND APPRECIATE THE TIME IT TOOK TO WRITE IT, PLEASE SELECT IT AS "BEST ANSWER." Thanks. The above is said without seeing your case file and without my understanding the entirety of the facts of your case. Depending on those facts, the above information be may incomplete or may be completely inaccurate. The above is intended as general information only based on what you described and not as legal advice. I advise you to consult with counsel who may be able to provide better information commensurate with a better understanding of your situation.
Being listed on the birth certificate or having a DNA test confirming you are the biological father, does not give you legal rights to the child barring a court order establishing you as the legal father of the child. In order to gain legal rights as the child's father you will need to file an action to establish paternity. If the mother is agreeable you could have a agreement prepared to address the issue of paternity as well as establishing a parenting plan, time sharing schedule and child support which could then be filed with the court. Without a legal establishment of paternity, you have no legal rights to the child and if the mother refuses to allow you to see the child, there is nothing you can do. I would advise that you contact an attorney to discuss establishing the paternity of the child which would then give you legal rights as a parent to petition the court for time sharing (visitation). It is best to establish your paternity sooner rather than later to avoid the mother removing the child from the State and then having a multi-jurisdictional issue when you try to establish paternity.