How do you get your son's father to relinquish his rights if you don't have an address for him All you know is that he lives in Long Island NY?
If you post in the newspaper that you are looking for him to see if he will do what you ask and he does not anwer within a specific amount of time, then you should be able to go ahead with your plans.
You can relinquish parental rights, but it does not absolve you of paying child support. When you relinquish your rights as a father, you are still a parent under the eyes of the law. You may also want to consider who's best being served by you removing yourself from your child's life just so as not to have to help raise him or her.