I'm sorry...I'm not a lawyer or a paralegal student, nor have I ever been in this type of situation personally and even I know the answer to your question! How do I know? FROM READING THIS FORUM! If you didn't find the answer to your question, you didn't look hard enough! As stated, this question has been asked and answered many times...at least the way you stated the original question. If you are looking for something different, you need to post a different question!
Just so you don't have to come back whining that you're being picked on, I'll answer your question...again, not as a lawyer, a paralegal, or any other kind of legal eagle - just someone who has learned from this forum.
No, you cannot sign away your parental rights just to get out of paying child support. You know why? Your second question answers it - because the courts would be jammed with deadbeat parents willing to sign away their rights in order to avoid paying support!
If Mommy doesn't like Daddy and wants her new squeeze to be "daddy," she has to have him adopt the child. The biological daddy will only be "allowed" to sign away his rights in a situation like that - where there is someone else ready, willing, and able to step up and take care of the child.
There...I have answered your question - I believe correctly.
You're welcome.