Well, I just got off the phone with tech support and they were actually very nice. We weren't able to fix the device but the guy I spoke with said (of course) that he'd never seen this problem before, but that it was probably a problem with the hardware and fully covered under the warranty, and he's sending out a one-day replacement.
Incidently, daffy4u, it's not possible to download the periodicals to the Kindle via USB - you can only get them via Whispernet. Obviously I can use Calibre, but that's a different version (arguably a better version, actually). But they gave me a few bucks discount because I'm missing a few days of the Times, so that's fine.
And as for that theory that the wireless should shut itself off if it doesn't get a signal - apparently (according the the tech rep) that's not true; the wireless will not turn itself off unless it is very low on battery. So that means if you're planning on being out of range for awhile you should definitely turn the wireless off manually since it drains the battery more to search for a signal than to maintain a signal. Or so I've heard.