This should be a link to the part you will wanna try first. I just dealt this this on my 2001 B5.5 passat.
Basically this is the relay that controls the blinkers, and the hazards. Its located directly behind the hazard button on the dash.
Just get some small screw drivers and carefully (it scratches) pry the button face out of the dash. Then using pliers (again carefully) pull the relay out. Its a really easy fix.
If changing this relay does not work, then you have a little more work on your hands. You might want to replace the turn signal lever. Its about an hour if your handy, and 2-3 if your not. I just did it on saturday. Involves taking off the airbag, steering wheel, and then the turn signal lever can be replaced. Costs about $75 for a VW OEM replacement.
Replacing both of those fixed my problem like a charm.