Problems with rear wiper on 2006 Avant

if it's the same as on the B6 A4 avant sounds like the seal inside the wiper motor has gone and the water from the washer is leaking into the motor, check if you have any water around the lock latch on the boot floor. Mine got worse when using the wash, wouldn't stop, parking anywhere but where it should and not working at all.
You can strip the motor down and try and reseal it but it won't last long so a new motor, about £100.

I just replaced the motor on my b7 Avant. was just all too easy to be honest, i wanted a bit of a challenge !!!!
€139, so about the 100 mark for you.
As Greeny233 was saying, make sure you check for leaks, the only 2 screws are in the pull down spaces on the left and right of the boot door, then the rest should just pull down. be careful of your boot lid light tho, dont forget to detatch it before pulling too hard !!

Dare i say, might just need to be taken down and given some WD40 ?