Files on third computer are not syncing with the files on my other two computers

Thank you very much for that, I learned a lot from your post.  However, it is now more interesting than ever.  Here's what I have learned:

Whereas I am a very experienced computer user (retired programmer), I know very little about Dropbox, and so have been experimenting with it and with Linux (I want to leave Microsoft entirely and create a comprehensive Linux solution). 

For background, you need to know my wife and I have different gmail accounts.  I have been using my password manager to log into Dropbox, using my gmail account.

What you sent me allowed me to discover that I had somehow unknowingly gotten logged into the misbehaving computer with my wife's email (even though I selected MY email when prompted by the Dropbox login process - curious.)  So I logged out and logged in with my gmail and verified that this was successful.

However, the problem persisted, but with a tiny but very important difference:  when viewing the files online, I saw (as before), the correct files (the ones synced on the other two computers), while (as before), in my Dropbox directory, the old incorrect files persisted (presumably having previously been synced with my wife's account).  BUT, the telling difference is that the little status icons were no longer displayed on the files, indicating to me that the local desktop process was no longer active and in a syncing mode of action.  Since I am unfamiliar with Dropbox, I can't speculate on why this is the case.