So I'm assuming you've found the usual sources, such as the USS James C. Owens website with reunion information on it? It looks like that's fairly active, last updated 3/5, and it might be worth contacting the folks listed on that page to start a conversation.

Our Guide on Military Identification may be of use to you. It provides a number of different resources, including how to request service records from a number of national agencies around the world, as well as graphical aids to assist in deciphering rank, unit, and other forms of badges or insignia. This could be helpful if you have photos or other memorabilia you need to identify.

The Naval History and Heritage Command has a Vietnam section, linked there, that could also provide a starting point for you.

Oh, also -- you could check in somewhere like r/Warships or r/navy to see if anyone there knows something about the Owens or other Sumner-class destroyers. Check with the r/navy mods first, maybe? but I think they allow that type of post.