Following up on this to provide an update, I tried as you suggested. Creating a GAM project with gam v6.10. worked, but when I ran
$ gam oauth create to create the oauth2.txt file with gam v6.22 using the new client_secrets and oauth2service JSON files, I ran into the same error as before:

I dug into this specific error more and I think I found the source of this problem - in February of 2022,
Google deprecated the OOB Oauth callbacks. it seems that when creating a new GAM project (at least in 6.10), and selecting Desktop App, it sets/uses that OOB value as a redirect_uri, which Google then blocks due to the deprecation. My original GAM project was a Desktop App project, which is why I selected that again.
So a GAM Desktop App cannot use the OOB redirect_uri which (older) GAM sets or expects, and a GAM Web App cannot set the OOB redirect_uri due to Google's deprecation. It seems like the correct fix for this is not the GAM version necessarily, but the type of application I make for a new GAM project. I think it needs to be a Web App and not a Desktop App, but with a different redirect_uri specified for the Oauth2 authorization callback. I found these migration guides from Google which should help me migrate and correctly create a new GAM project Web App:
I'll sync back here after I go through those to figure out the new configuration. And I apologize if this is too much information, but I want to respect your time by staying in sync with you on this since I'm sure you have plenty to do already, and I really appreciate your help.