I got help from our IT dept and it solved my issue. Here's what they had me do...
Quit Mail
Open System Preferences>Apple ID>iCloud Drive - click Options
*Important - you want Mail under iCloud Drive Options, NOT "iCloud Mail" from the main menu
Uncheck Mail
Open Mail and add/adjust signatures
Quit Mail to save changes
Open Mail again to make sure signatures saved, then quit Mail again
Open System Preferences again >Apple ID>iCloud Drive - click Options
Recheck Mail
Open Mail to confirm changes saved
After many other suggestions tried, this one actually worked. I also noticed that some of my rules had disappeared so while I was adding my signatures back in, I entered those as well and everything saved.
Hope this helps someone else!