The dad cannot just "sign his rights over" and avoid child support. He cannot relinquish parental rights unless it is in connection with an adoption proceeding, or a juvenile court case to terminate parental rights for abuse or neglect.
DISCLAIMER: This answer is provided for general educational purposes only. By using or participating in this site you agree and understand that there is no attorney client privilege between you and the attorney responding. The law changes frequently, and varies from jurisdiction to jurisdiction. The information provided in this answer is general in nature, may not apply to the factual circumstances described in your question, and may be different in the State where the facts occurred. For a definitive answer you should seek legal advice from an attorney who (1) is licensed to practice in the state which has jurisdiction; (2) has experience in the area of law you are asking about, and (3) has been retained as your attorney.
Sign up to receive a 3-part series of useful information and advice about child custody law.