Can the father of my child sign over his rights to our unborn child?

This depends on context and what is meant. Dad can relinquish all parenting rights to Mom but that can be undone or modified upon consent or a showing of changed circumstances, and doesn't extinguish Dad's support obligations, *unless* there is a stepparent adoption or judgment terminating parental rights.

He can sign over his custodial rights to you, but neither of you may sign away your child's right to be financially supported by both parents to the best of their abilities, unless there is another man willing to adopt the child and take over his financial responsibilities for the rest of the child's life. There is a public policy that children should be supported by both of their parents, and that if (for any reason, at any time) the custodial parent needs help from the state (food stamps, state aid, medi-cal, welfare), that the state should be able to make sure the other parent is providing what they can financially, and it does not remain just the state's responsibility. You can agree to give you full custody and to set child support to zero for now (as long as you are not receiving any aid).