Hello cbakl,
Thanks for posting here on the Apple Support Communities! This is a place where other Apple users can submit questions or respond to others to help lend a hand. If we understand correctly, you've tried the guidance from If Mail on your Mac keeps asking for your password - Apple Support and the same continues. Great troubleshooting so far.
Since the same keeps happening, it will be helpful to isolate where this happens. Does it happen in safe mode? Access safe mode using the guidance in How to use safe mode on your Mac - Apple Support.
Testing in another admin user account can also help to isolate where this occurs. If you don't have another admin user account to test, you can set up a temporary account using the steps from Set up users, guests, and groups on Mac.
In most cases, issues that occur in one user account are related to login items. So, if you find that an issue only occurs in your user account, remove your login items and test again. Check out Remove login items to resolve startup problems on your Mac.
Take care.