Am I legally obligated to give the baby my husband’s last name?

The child should have the name of the father on the birth certificate. If you are married, there is a presumption that the husband is the father. Of course, if the bio dad wants visitation, he can petition to determine paternity and have his name on the birth certificate.

Of the child is born during the marriage the your husband is the presumptive father and you can give the child the family name.

No, there is no presumption that a child born to a married couple bear either parent's surname. Which means you could actually give the child a surname that neither you nor your boyfriend have.