***UPDATE*** Return complete!!!!
Short version: It took awhile, but the return is complete and I was rewarded!
Long version: I contacted the email on the school's alumni website from the link Aquaman posted. I told them to pass my information along to the original owner and have them contact me. The person was a bit recalcitrant and wanted me to send her the ring to give back "for privacy reasons." I understand that, but wasn't going to take that risk and after a few back and forth emails I got one from the original owner. A very nice lady.
I sent her the pictures to indicate I was not scamming her and told her she could have it back if she sent me the $11.85 in shipping. I was happy to give it back, I just didn't want to be out 12 bucks to do it. During our many emails, I told her about this message board thread and she even came here to read it and was amazed. We talked about the island and how we stay at the same resort and where we eat, etc. Somewhere in telling her about how I had it in the envelope to be melted, I asked her to consider reimbursing me for the value of the gold (after she got it back of course).
I sent it and followed the tracking all the way, excited for her to get it. Once it was in her hands, she sent me the picture I asked for (see below). Much to my surprise, a few days later she Venmo'd me money equal to it's gold value. I am quite sure that both of us are very happy with the return of this ring.
Shout out to Aquaman, it wouldn't have happened without him. I will happily buy dinner if we ever cross paths.