If the card is a Built by ATI card, you have a 3 year warranty on it. Yes. I said 3 year warranty.
Definetly check your system ram. run memtest86+, try reseating the ram (make sure your are grounded) in case you have jossled your chassis lately.
The other thing to check is to ensure that the external power adaptor is connected properly to the card and that the card is seated properly in the slot. Lack of power will prevent you from getting 3D accelleration and an improperly seated card may slow the system down or potentially cause shorts between the traces.
A friend of mine had the exact same problem with his 9800 Pro. Artifacts look almost identical. We troubleshooted it one weekend and the end result was a lot of tinkering and reinstalling with minimal results. Best we could do was getting it to work ok for 10-15 minutes however the artifacts would come back. Bottom line, he ended up using an old card I had and the computer worked fine. Sad to say this may indeed be the end for the card.
I've had the same problem with my card. I ended up replacing my power supply because the 5v line dropping to low. It starts with the graphics card, and then the computer would start experiencing other strange problems.