Define damage and burn marks (picture?) Don't be offended, but are you sure you're getting the right bulbs and installing them correctly? I ask because when i think of burn marks, I think of shorting, maybe loose bulbs, moisture, bulb installed backwards or damaged socket. The relay can cause your bulbs to stay illuminated, but I don't know why it would cause the burn marks.
My guess is the bulbs are shorting out, although I'm not sure how that is happening. Somehow the bulbs are being back-fed power causing them to stay on.
It could be you have 2 problems, a bad relay and bad bulb (taillight) clusters. The cluster causing the relay to fail.
'00 A6 Avant Quat 190k
50% Front 20% rear Tint
Euro Blindspot Mirrors / LED interior lighting