I am an unmarried man in the state of New York. My ex is pregnant, and I believe the child is mine. She wants me to sign papers to give the child away, and threatens to have someone "pose"as me and do it if I refuse. I have no idea what my rights are in this situation--and she is due any day now.

Well of course tlil the child is born you can't do anything. But you can start getting paternity paperwork filled out to be served when the child is born. Once born you can though DNA prove you are the father.
At that point you can request visits with the child and arrange the child support you will owe.