For anyone following this thread or who might read it in the future. I opened a support ticket (just no help for it) and this is what we came up with: some service (could be SM, could be some AV app, could be a system app; we never knew) was locking the mailinglist.db files. Once those files are locked, they show as 0 KB in Explorer and they can't be accessed. When the SM admin interface can't access them, you get the Oops! message and the page freezes until it times out. Kind of ugly way to handle the error, but there you have it.
The got an unlock when I did a system update followed by a reboot (being a mail server, we don't reboot very often). Once the .db files were no longer locked, the problems went away.
The really mysterious part of this was that when one particular mailinglist.db files was unlocked, it didn't return as its current version. Instead, it came back as a version from 2012. That is, there is one particular domain that adds new mailing lists on a semi-annual basis (two from 2010, two from 2011, etc.). Before the .db file got locked, that domain had the two most recent mailing lists from 2016 (plus two from all the prior years). Then the file was locked and mailing lists in that domain did not work at all. Finally, after the file was unlocked, mailing lists worked again but there were only lists from 2012 and prior. Also, other posts-2012 data (such as a new user field and changes to subscriptions) were missing. I have no idea how the system put back an old version of the file; I wouldn't know where to find one if I wanted it. Fortunately, i had a recent backup and restored the .db file from that. So now the .db file is up-to-date and the system is working.
Tech supporting also recommended upgrading to version 14.x or 15.x (we're running 13.x) because those handle filing locking much better. So that's next on my agenda. I was really hoping v16 would come out and we could skip straight to that, but I don't know when that will be.
Thanks for all your help.