How does a father relinquish parental rights in the state of Washington

What do you hope to gain by relinquishing your rights?
Even if you could you would still be held accountable for child support.

The mother gets remarried and her husband adopts the child.

Well, first the father finds the mother a suitable husband. Then after the mother and new husband are married for a year (sometimes longer depending on the state) asks if the new husband wants to adopt the child. If the new husband does then they get a lawyer to draw up the papers and it all goes smooth as silk from there.