Hey....I have an 02 J TDI too!! Yer a good man! ..ha ha
Here are some things I would investigate. I'll dole these out in order or easy to hard:
1) Do a battery load test: Get a cheap heater style load-tester from H/F and make sure your battery is good. Do this before the others b/c if it's not, you'd be wasting a lot of time. These trucks get wonky when the battery's fail. Even if it test fine with a VOM, they can still fail a *load* test. I've had it happen on my own Ram. Or take the battery out and to OReilly/Autozone/Advance/Napa, and let them test it...for free. Obviously make sure your battery grounds are good and all that. Like I say, it's worth it to cross the battery off the list of being good/bad before you go down the road too much. Even if the battery isn't very old.
2) Check/clean the major grounds: You already found the ground/s at the battery cables. There's the metal strap (positive) at the fuse box). But.... Above the driver's side front tire behind a plastic door, and back behind the fuel door by the filler neck area...there should be several Gnds there. These can get real rusty in the North. Take them apart and make sure they're good and clean if yours look trerrible. Again, there's one back by the fuel filler neck.
You might even want to add a wire from the Neg battery cable to the firewall....just to see if that helps anything. Or better yet, get a real ground strap from autozone. Since electron exchange occurs mostly on the exterior of wires, a fine-braded cable will work a lot better as a ground. That's why they use fine wires in cars. And like you say, make sure the ground to the engine and engine to frame/firewall are all good.
3) Corroded TIPM Connectors: If none of the ground fixes helped, ....The connectors under the TIPM can get corroded. That can cause wierd things. In fact there have been some owners in the NE who've had many wires at the TIPM connectors corrode back into the wires ..and needed to splice in a new connector (think of all the wires there!!). They had a bunch of odd gremlins too. So you might do a search for TIPM Computer Connectors...or something like that. You might dig down and look under the TIPM, disassemble a bit to the computer and inspect the connectors. There are videos on this. If you're lucky it's just corrosion and you can eat the corrosion away with some phosphoric acid (Menards) and put oil on them once they are clean of corrosion ....and be good after that. If that was the root problem..then it would be a pretty easy fix.
4) Rodents: Yeah that was good advice up above. This time of yr rodents (that includes squirrels) are looking for a place to go. And the harness is only half taped up!! There are a lot of places where wiring is exposed to munch on. If you're a hungry rodent, maybe a little strung-out from the cold, or just feel the need to work yer fangs down a bit...what could be better than a bunch of candy-colored wires!!! Ha ha. And maybe those wires are under the computer too. So go along the harness and very carefully inspecting things is another good idea.
5) It could be separate components too...and the shift control knobs do flake out. That's another possibility. Obviously if you don't find anything above, consider ordering a new knob.
Keep on Jetta-ing!!
