I FIXED IT MYSELF. I just updated to macOS Monterey 12.3.1 today, April 12, 2022. After the update, I was unable to save my AI files to my external HD without receiving a message that "the file was open in another application and could not be saved" or that I did not have priveleges to save the file as it was "read only."
Quit out of your programs.
Go to System Preferences
Click Security & Privacy
Click the Privacy tab
Click the lock to make changes and enter your password if this is protected
On the left, scroll down to Full Disk Access
On the right, make sure that AI, Photoshop, Indesign, and any other promgrams you use are CHECKED
Click the lock to keep changes from being made
Restart your programs.
I tested this numerous times and it seems to be resolved. Nothing as effective as an angry graphic designer with no IT dept.