When my son(?) was born in New York I signed a form at the hospital stating that I was the father, but now have some doubts. Can I still request that the court,in North Carolina where I live, to have a DNA test for establishing paternity.

Depends on where the child lives. You would have to challenge paternity in the jurisdiction over the child/mother.

The other question is how old is the child now? Many states have a fixed age at which paternity can be challenged.