What can I do to correct being awarded a NAM medal that never made it into my service jacket?

I think you'll need to file a DD-149 to get a issued a DD-215 to amend the DD-214

What medal are you trying to get added? What unit and exact dates of service in country and we can pull up the unit command chronology that will show where the unit was when you were with them. This then can be linked to your SRB that has all your dates and unit assignments. You might even have to track down some officer you served with who can verify your request. A record of the medal will exist somewhere in there.

Reading in context, are you referring to a Navy Achievement Medal? If the award was approved, the issuing command should have published orders awarding the decoration. If you received copies of the orders, simply present a copy to your current records manager. If orders were never published, there still should be a record of the nomination for the award and status of approval/disapproval. The awarding command should have that information.