I HAD to come back here and update. So many others DON'T do that! They don't explain what finally happened!
So, after trying out a LOT of variables, I finally filed a trouble ticket (marked "not urgent") and they replied in a couple of hours tops.
I do NOT quite understand what the problem was, but it was explained that, after my server had been scheduled for migration to a new server, my domains were held back because I still had a couple of them set to an older version of PHP.
So, all my domains were on the same server, nothing had changed, yet SOMEHOW(?) their performance changed? I still don't get why that would have made any difference, but it sure got my attention!
The tech explained that all I'd have to do was to switch the PHP version to a higher level, and they could make the move to the new server. In a few hours, it was all done!
The new server performance is noticeably faster than it was before the slow down!
So color I'm very happy!
BTW: Changing the PHP level didn't "break" anything.