If I give a child my name because I married the mother but then divorced her, am I obligated to pay child support?

Yes. A child born born during your marriage is presumed to be yours. If there is a support order, there is a finding in a court order that you are the father. The fact that you do not have a relationship with the child is not relevant to your obligation to pay support. It would be very difficult to now argue that you are not the dad. You had that chance many years ago and apparently chose not to do so.

You have an "estoppel" problem. (Google that word for a legal definition.) It appears from your description of the problem that when you divorced the child's mother, you didn't disavow paternity. It appears that you let a divorce decree happen stating that you are the father. The court isn't so much interested in "you," but instead is more interested in the child's interests, and in the state's interests, in this situation. Both the child and the state need your child support check! Divorce courts are referred to as "courts of equity," and it is said that: "Equity aids the vigilant; not those who slumber on their rights."