First I would try removing the heatsink, applying paste, and re-seating it. As its used, I assume it was shipped to you and rough enough handling could have broken the chip to heatsink "bond". Without a heatsink, a beefy gfx card could be instantly overheating.
I had a card do the same to me years ago and I brought it back to life with the old bake in the oven trick. Risky of course, and be very careful, but if nothing else works: http://lifehacker.com/5823227/save-dying-video-cards-with-a-quick-bake-in-the-oven
Make sure ALL plastic is off the card. It will probably smell a little burny, but that is normal. Make sure your oven doesn't under-report its temperature and turn the oven off immediately if you see any hint of smoke. I wouldn't do more than 10 minutes either.
Once its done and fully cooled off (couple hours - let it cool slowly so nothing cracks), make sure to use a quality thermal paste when putting the heatsink assembly back on.