I'm not entirely sure how one switches labels on a battery...
If you smelled cables burning, it might just be that. Sounds like you'll have to run through the whole wiring system for audio. That being said, are you sure it is the speakers causing the problem? Do you have another headunit you can swap in, or maybe even plug the speakers into an amp to test them?
My strategy would be that: test each individual component until you find the broken one, and replace. I hope you have a multimeter
Where are you located? I might have some time in December to help you out.