For anyone facing this same thing: We finally happened upon this page. I'm not sure it actually explains that even though Shadow Copy is DISABLED the settings in the below dialog STILL AFFECT what Windows Server Backup does, which uses shadow copy to create its backups.
Right click the drive, select "Configure Shadow Copies...," select your drive from the list and hit the "Settings..." button to get this:
This shows the errant settings for drive M:, which is a fairly new drive. We don't know what or why something decided that drive M:'s shadow copies should be stored on drive D:. Maybe because Windows Server Backup was already using and backing up drive D: or it saw that there was far more space on drive D:.
When we changed the settings as described below for drive D: nothing happened, that's because drive D:'s shadow copies weren't the problem. On a whim we checked drive M:'s settings finding it was storing on D:!!
To fix the problem set the value highlighted above to the minimum of 320 MB, come out of all dialogs and keep checking your drive space. Once it goes down you can reset the limit to something more reasonable.
It appears that once something goes awry with Windows Server Backup it starts "orphaning" shadow copy files in the System Volume Information folder. We had files in there for everyday going back 7 months, even though the backup appeared to be running correctly. I have not taken the time to slog through Event Viewer logs to see what might have happened on that day. The first backup that ran after making this fix has NOT started piling files in the System Volume Information folder again, so that's good enough for now.
If anyone has any insight as to what sends Windows Server Backup into save-stuff mode I'd be very interested.