How do I give up legal custody of my son to join the Air Force?

Sorry, but that's not how it works. Once you have a kid, you are responsible-at least financially-until that child is 18. You can't just sign a paper to wash it away. He's not a puppy that you can re-home. Your only way of being absolved of responsibility would be if you ex has a partner that wishes to legally adopt your son as his/her own. Barring this, you will be held accountable for helping to financially support him and he will remain legally yours. As far as paternity goes, it really doesn't matter what is on the birth certificate. If you stop paying child support, your ex has the right to have the courts subpoena a paternity test and you will be legally responsible for all back and future child support.