After talking to five (!!!) different people in Apple support from Sunday through Tuesday, I finally convinced one of them that yes, there was a problem with iCloud asking for my password repeatedly. Nobody up to that point seemed to believe me, even though I'd taken a screen shot of the annoying message, etc. They wanted to see the message actually pop up while online remotely with me.
Finally, after a bit of begging on my part (thanks, Apple, for the great customer service) a support person suggested that I reload the latest version of El Capitain in recovery mode. She walked me through it (took about an hour), and so far for the rest of yesterday and today, I'm no longer getting the annoying request from iCloud. I believe that there must be a software glitch somewhere in the update, but reloading it seemed to fix this issue for me.