Do both parents have to sign.. say if my sons father wants to sign rights over to me. Do I have to sign also?

If the judge would even allow it, yes both parents and the judge would have to sign,

They almost never allow a parent to sign over their rights, unless the custody parent is remarried and they want to adopt.

Also with the exception of a step parent adoption, if the court would allow one parent to sign over their rights, normally they still have to pay child support.

One parent can not just sign over their rights to get out of child support