I just found out I have a child. Can I just sign over my rights and avoid child support?

It is very unlikely that “signing over your rights” would be approved by the court. The issue you bring up is the termination of the father’s parental rights. Dallas area judges are quite reluctant to allow a termination of your rights unless someone is also seeking adoption of the child. It is more likely that the paternity of the child would be established at this time. Texas law recognizes that children have a right to know their parents, and vice versa.