SOLVED

Alright I have no idea how this fixed anything at all, and I pity anyone who has this problem themselves. To fix this, I pulled a controller out of my computer's USB slot. Not even a controller, a Mayflash Gamecube Adapter for plugging in controllers. As soon as I did that, all the sound came back within my active session of FFXIV. I don't even want to know how this fixed anything, but if you're having the same problem, try pulling out USB devices.