[Got worse] Weird Nexus 7 (2012 3G) charging problem, ie. it thinks it isn't

Edit: It actually started misbehaving again, and now won't even turn on; plugging it in results in brief flashes of the backlight turning on with a scattered few lines or so on the screen.

So, ever since upgrading to 4.3 (although it's remotely possible this is unrelated) my Nexus 7 has been really flakey about charging. When I plug it in, it'll stay at whatever charge it's at . . . which for example, at the moment, is 0%. It doesn't appear to detect that it's charging. However, if I plug in headphones I can hear that it's at least getting power, and it sure as hell lasts longer than you'd expect something at 0% to do.

At least if I turn it off, I get an indication that it's charging, as that simplistic white-on-black charging graphic appears. But it always shows that it's starting from 0, and never appears to gain any ground.

I've managed to get it 'charged' once or twice since the 4.3 upgrade, but again, it does seem to be charging regardless, as I can take it off the charger for a non-zero amount of time and watch 720p video streaming over my local network and hey, it doesn't just immediately turn off.

Behold, a screenshot from moments after I plugged in the charging cable---yup, I'm using the original one, and in fact the screen turned itself back on when I plugged it in, it just still thinks I'm not charging it for some reason:

Screenshot_2013-08-20-12-11-50_medium

via lh3.googleusercontent.com

Has anyone else encountered anything like this? Does anyone have any suggestions on what I could try? (And no, "buy a new Nexus 7" isn't valid...because the cellular version of the 2013 one isn't out yet :P)