The OCGA 19-12-1 states that if the location of the other parent is unknown, you can serve them your petition to change a minor's name via publication in the local newspaper. It also state that written consent is needed.
How can one obtain written consent when the whereabouts of the other parent is unknown, and what happens after the 30 days of notice when no consent is received?

Has anyone experienced this or has any information to guide me?

You are misreading the statute. If you follow the rules for publication of notice of the petition and no objection is made in court then the change will be granted.