What’s the best way for my husband to legally adopt my child?

You can do a step parent adoption.  If the biological father does not voluntarily sign a consent to the adoption, his parental rights can be terminated involuntarily for failure to maintain a meaningful relationship with the child for at least 12 of the past 14 months; or terminated for failure to provide support to the child for the same time period.  If a process server cannot find the bio dad, then you can ask the court for permission to serve him by publication in the newspaper.  I can help you complete the process when you are ready to proceed.