Thanks to the Drupal Dude, who came through again with flying colors, I got the upload directory schema sorted out, have tarred-and-transferred all my documents, and have just spent the last hour cheerfully linking documents into the existing nodes on the new site. Cross that puppy off the list! w00t!
What I did wrong in the IMCE settings is this: I missed that it's all relative to the main directory "files". By setting "files" again as a shared directory for authenticated users, with subdirectories of blah blah blah, the system was going "Oh, you want that for each authenticated user". Turns out if you put "/" into the shared directory setting and leave all the rest blank, voila, now the proper directories are visible, navigable, and the files in them are linkable, to all authenticated users. Hurrah!
Sadly, I must restrict myself now - half a day on Drupal, half a day on reimplementing our EDMS. I wanna keep cranking on the Drupal pages - linking in documents is a good half of what's left to do, and I'd love to be able to mark those pages as finished! But, all in good time...
It's a lovely sunny 70°. Wish I could send some of this to either coast - have family on both sides that're getting smashed by Mother Nature right now.