Legal Question in Family Law in Virginia

You don't simply give up or terminate your parental rights to a child by signing

a piece of paper and having it notarized. Such a matter in the Commonwealth can only be decided by a court of proper jurisdiction after the issue has been placed before it through a valid petition for adoption or other matter necessarily involving the termination of your parental rights to this child.

And, no, once such parental rights are terminated, you would no longer be obligated to pay support for the child.