More work has yielded a little bit more information, but not a solution yet.
When I replace all of the images with the new files with FTP, resized appropriately, and empty the cache on my browser, the images behave as expected within posts and pages, and the new smaller file pops up when clicked rather than a cached version of the giant image.
However, the Media Library is still looking for the giant file, because it has stored the dimensions in the database. So on the "Edit Media" page in the dashboard, the image is replaced with a broken icon. Same with the "View Attachment Page" - broken icon - even though the file does exist with that exact file name on the server, Wordpress can't find it to serve it up.
Clicking "Edit Image" from the "Edit Media" page does bring up the new file, and if I edit it there, the database records the new dimensions and everything is rosy. Same if I use "Enable Media Replace," and point to the new smaller file.
But I have 689 images to deal with, so going in to every single individual entry in the Media Library just isn't feasible.
Any suggestions, folks? Please?? There has to be a MySql query or something that I can do to blank out the dimensions, right? But I'm not all that versed in MySql, so I don't want to poke around and end up destroying something inadvertently.