Hold the fort....I think I got it sorted out. FINALLY.
In case anyone else runs into this, here's what I finally figured out:
The site uses the Modernize theme. It has Really Simple Captcha built in. I added Contact Form 7. It worked, so I didn't give it much thought.
I went into the themes functions.php file and commented out the function that added Really Simple Captcha. Then I installed the RSC plugin separately. Voila!
On the advice of some forum info, I'd tried to install RSC but since it wouldn't do it (gave me an error) I changed the plugin name on install. Fine, but Contact Form 7 was still trying to work with the pre-installed version which apparently wasn't finding the right directory...or something.
Anyway, I finally think it's fixed. Thanks all the help trying to get it sorted.
Whew!
I'm marking it resolved and will let you know if anything crops up.