Not in privacy mode. Website data is not cleared except when the browser is restarted. Disk space is fine. I doubt it’s an OS bug because the extension used to keep its settings and I haven’t updated macOS since. The only things that have changed are regular Firefox updates, and the extension updates, one of which apparently caused the settings to stop being retained.
I whitelisted foe-rechner.de for cookies and site data. It seems to be working now, but will need to test for a day or two.
about:preferences#privacy > Cookies and Site Data > Manage Exceptions > foe-rechner.de (Allow)
about:preferences#privacy > Cookies and Site Data > Manage Exceptions > forgeofempires servers/worlds (Allow)
Thanks for the reply, andrgin.