Can a mother sign over her rights to the father at the hospital for the birth so father can take home baby?

You can allow dad to take the baby home from the hospital, but the hospital does not have any paperwork for you to sign over your parental rights. However, you can file a Suit Affecting the Parent-Child Relationship and ask the court for an order appointing dad as the sole managing conservator. The court probably won't allow you to terminate your parental rights unless the baby's father is married and his wife wants to do a stepparent adoption. You can certainly get an order giving the dad most of the rights concerning the child. You will still have some rights and you will still be the child's mother, but the dad will have most of the rights.