Do I still have to pay child support if I give my rights up?

The only way to stop paying child support is if your rights are terminated by the courts... and that's not an easy thing to do... typically, there would have to be someone willing to adopt the child/ren to take over your role as their parent.

No court is going to grant a TPR just to allow a parent to get out of child support. Try researching the hundreds of threads in the Family Lawy forum (where I moved this to) that ask the same question.