Update:
Last night I suspected defines.php might have been corrupted, but this morning, it seems to be working okay.
There have been no recent changes on the server end that would explain this.
I'm thinking there might be a conflict with server-side caching between settings in Joomla and the server? The problem must be on the server since it malfunctions regardless of browser or computer.
Can't really do much to troubleshoot it now that it's working. If it starts acting up again I'm going to also have someone else try logging in from outside the LAN just to see if it has anything to do with the IP. I have had problems like this before caused by logging in and out a lot from multiple sessions, different ID's etc.
I do not now think this is a bug.