OK, things seemed to be working out for a while after the server move but then, as I was uploading the last of the 20 or so clips, I not only got the “connection reset” message, but all the files I’d uploaded disappeared, both from the server and from the WP menu.
Not only that, but the pages I changed in the last couple of hours rolled back to their previous versions, as if I hadn’t published them with the changes (the most recent autosaves are still there, without the clips).