I got it working. I'll review what I did here in case somebody sees this in the future.
When the problem first happened I simply removed and re-installed FMPA. Upon doing so, the install wizard asked me to reboot my computer so that it could finish doing it's thing, which I thought was a little odd because that doesn't normally happen with FMPA installs in my experience. After rebooting, though, the problem still remained. That's when I posted this thread here.
Between the time I originally posted and the time I got a response I was messing with it more and got it fixed (but forgot to come back here and update.)
All I did differently was this time I did another uninstall, then I manually removed the leftofer FileMaker folder under C:/Program Files (x86). I also manually removed the registry entries I saw for FM, and then I rebooted the computer.
After that, I did another install, and this time it did not ask me to reboot. Everything is working fine now.
Still not sure what caused the issue but at least I'm back to a usable state.