However, the problem suddenly did go away for two days, following a recent MS-Windows update (December 11). But that also led to four instances of the "Critical Process Died" stop code. When I undid the update (which stopped the "Critical Process" code), the "save" problem returned. One thing I have noticed is that if I keep a Word document open for less than 30 seconds, I am not asked to save before closing. Only documents that are open for more than 30 seconds prompt me to save, even if I have not made any changes.
It may take Word a bit of time, especially with longer documents, to update fields in the document. With a copy of your problem document, try selecting all (Ctrl+A) followed by Ctrl+F11. This locks all fields in the body of the document. Save that copy under a different name.
When you reopen it, do you have the same problem when you close it without having made any changes?
It would help if you could save a sample document without confidential or proprietary information on OneDrive or DropBox and post a share link here. Here are instructions on how to do this: https://addbalance.com/articles/Why%20a%20sample%20file%20is%20important%20for%20troubleshooting.%20How%20to%20do%20it.htm