Messages and FaceTime repeatedly asking me to log in.

The Messages and FaceTime apps are both asking me to log in over and over. When I do so, they seem like they're working for about 10 second, then it goes back to the login prompt. If I cancel the login window, it quits the app, and that brief period between prompts isn't quite long enough to get into either app's preferences to make any changes.

This is not affecting anything else as far as I can tell. Mail, Calendar, iCould Drive, and the App Store are all working fine.

This is happening across at least 2 Macs. My M4 MBP and my M2 Mini.

My iOS devices (Phone, Watch, and iPad) are all working as expected with Messages and FaceTime.

The only thing that I changed, which may or may not be related, was setting an old iMac (running Mojave) up for my Dad to use. I have an account set up on it as well as the admin which is connected to my appleid.

I have removed that computer from my Devices in the account section of the system settings. Which did not solve the issue.

I have also tried disabling Messages in the iCloud settings and then reenabling it. Still no luck.

I'm quite baffled. Any help would be greatly appreciated.

[Edited by Moderator]