The bayonet-ended blinkers in the backbox were working fine but I guess when I opened the backbox earlier I jostled them "just right", because now they're not turning on. If I press on them hard and fiddle with them a bit, they'll work - but as soon as I let go they stop. Any tips on fixing - do the seats just need a good cleaning or something?

OK so I actually discovered something new here. I had blinking bulbs in the bayonet sockets, and when I tried it with non-blinking bulbs they lit just fine. I also discovered a couple of the blinking bulbs had gone bad, in spite of being only a few weeks old.

So here we go:
I had blinking bulbs in the bayonet sockets for about a month and they worked just fine, until they didn't.
When I removed ~1–2 of the blinking bulbs, the others started working just fine.
Right now I swapped 1 blinking bulb for 1 regular bulb and replaced the bad bulbs, and now all is good.

But I have no idea why having several blinking bulbs in would work fine for a while, and then just...not work. No fuses blown, no random resets, no problems at all that I've seen, other than the bulbs.

Yes, using LEDs. I did poke around at the wires behind just to make sure everything was connected and to see if jiggling it helped. I'll take a deeper look and try your suggestions. Thanks!

Just wanted to post my resolution -

Wires to the lights were fine, terminals were clean, etc. I ended up finding two problems:
1) One of my lights had burned out! Even though it was a brand new bulb. Didn't expect that, these things happen. I discovered it when my wife suggested putting the bulb somewhere else (because she wanted to see what it would look like).
2) Turns out the socket I was most worried about is "fine" - one of the arms on the back of the socket is slightly loose, so when monkeying with the bulb, it turned the socket against the arm in such a way that the connection stopped. When I moved the arm straight like it's supposed to go (parallel with the opposing arm) the light works flawlessly.