sound to me like there is a possibility there could be a damaged wire in the door hinge.
I just recently fixed this exact issue on my Uncles 2003 Audi A4 B6 1.8T. Some of the wiring was physically cut in two where others were merely making contact in certain angles the cables where.
Cut and rejoined all wires. Everything works perfectly.
I may suggested check those wires first before door removal. As it's a lot less hassle then.
hope this might help.
Keep us posted to what you the outcome is.
I had exactly the same on my Mk3. When you think about it, the gubbins in the door doesn't move (well, not within the door anyway - you know what I mean!).
Whereas the wire leading to the door is forever getting bent through almost ninety degrees. It wasn't hard to get at or fix.
HOWEVER -
It's unlikely that both front and back would go at the same time, isn't it? Perhaps you should be looking behind the head unit.