can I sign over all my rights to the mother of my children if the mother and I both agree?
It is rare when the state will allow a parent to sign away (relinquish) his or her rights.
Even when you are allowed to relinquish your parental rights, you don't extinguish your parental obligations.
You are still required to pay child support, especially child support that is in arrears.
If you and the mother were NEVER married, you never went to court to establish your paternal rights, you have NOTHING to relinquish.
You would have NO parental rights.
Yes, that would be true if someone placed your name on the child's birth certificate.
However, no court is going to be swayed by some dumb guy on the internet and his dumb opinion. YEAH, that is me, buddy, the dumb, unknown guy on the 'net.
I suggest you consult with a lawyer, because you'll need one if you wish to pursue relinquishing your parental rights, assuming you have any.