I really appreciate everyone's responses! It's nice to know there are others trying to solve this also.
We are not using DFS. We only have one office location, one central file server (and it's a Windows network share). We are using a mapped letter drive (we all use the same letter). We aren't working remotely or accessing these dwg files via cloud-based storage.
We also aren't using any of the other things @PaulTOB mentioned (NAS, Linux server farm) - but I agree 100% with @PaulTOB's reasoning and that with those would come issues as described.
With the balloon xref notification issue.. At first we thought it was giving out random usernames. But it became a lot easier for us to pinpoint after one notification it gave us said a username of someone who was longer even working at our company. That's when we realized it was reporting the username of the person who saved it 2 times ago, not the most recent saver's username. Then it became clear why it appeared to be working sometimes and not others. Because often you're saving multiple times while you're working on it. So after that first balloon notification goes out with the previous saver's username, then the next time the balloon goes out it will say your username, even though it's still broken. But once another user goes into that xref and saves...the problem starts over.