What does giving up parental rights maen in Texas? Is the custodial parent still responsible for providinf health insurance, child support, etc if they give up right?

If a parents rights are terminated it means they have no rights to see the child or rights in how the child is raised. If a parent gives up rights to another for adoption it means it means they have no obligations to the child, rights to see the child or rights in how the child is raised.

Yes, the non custodial parent can still be held responsible for health insurance and child support