This is my first time using the "Answer your own question" feature. I hope I'm doing this right. My Title triggered a warning that my question looks subjective and will probably be deleted.
I searched the site and didn't find any questions that matched the level of detail that I put into my response below, so I'm just trying to help out some fellow programmers by posting this.
As the administrator of a Google Apps domain, how do you use the Google Email Settings API with OAuth 2 to programmatically set the email signatures of users on your domain in Google Apps Script?
I experienced some confusion when trying to get this to work after OAuth 1 was deprecated, but with some help from awesome SO users, I was able to figure out a working solution.
First, you need to follow the steps to add this library to your Apps Script project:
https://github.com/googlesamples/apps-script-oauth2
After you have that set up, you can use their library to create an OAuth 2 service that is needed when calling the Email Settings API. Here is my working code: