Can the state hit me for child support if the mother if my alleged baby adopted her out to a relative without my consent?

The mother of my alleged baby wouldn't let me sign the birth certificate and isn't cooperating with giving me a dna test. She asks me for money and I'm not even sure it goes to the baby. Her actions and the fact that i don't know her very well and we slept together once make me doubt I'm the father but if I am, I want to he a part of the kid's life. She had problems with Department of Children and Family Services, and had her step sister adopt the baby to avoid her being put in foster care. She did this all behind my back. They are hard to reach by phone and move often which makes it hard for me to keep in contact. I am a college student in a nursing program. I don't have much money so, can I petition for a free dna test since she keeps asking me for money? If I am the father, will the state come after me for child support? The adopted parent is single and on welfare and the baby has sickle cell anemia. She claims she didn't want me to be on the birth certificate so she could get more government assistance since neither of us has any money. I'm just really confused about what to do. I don't want the state to kill me with child support fees because I barely make $500 a month and the child is 2 but if she is mine I would like to know for sure and know what my rights are. With me being a full-time student, I don't think I can afford a lawyer or child support or full custody, but I want to know if that is in fact my baby so I can have a relationship and provide everything I can.