I did another clearing of cache not long before your test (thanks very much for doing those) and I haven't seen the problem since.
The database errors came up when I had upgraded to 6.11 earlier today, but seem to have stopped after I then did the 6.12 upgrade.
So, right now, fingers crossed, it looks like things may be working OK. I will see what happens more more uses start using it now that I've told them the upgrades are done.
Many thanks for the help and the promptness.
Oh - any idea why RECOVERY MODE didn't work? I put the line as the first entry in the constants.php file as shown in another post.
Oh, I also had run the background job to complete indexing.
