Sounds like the card, likely one or more of the on-board memory chips failing which will cause artifacts. Sub-par performance by the PS is another cause. I did have a failing mb cause instability at start up, but the problem was not manifested as artifacts, just intermittent BSODs or system crashes.
You could try the video card in another AGP system to see if the problem is duplicated there - maybe a friends. Does your old 6600Gt still work?
What drivers are you using? From my experience, Cat 7.7s are about as far as you can go with most X1950 AGPs and maintain reliability and stability. Maybe someone reading this post and who has tried to run Crysis with an AGP X1950 Pro can share with us his/or experience. If you've read through many of the forums today, it is obvious that Cryis is a demon when it comes to stress on video cards and indications are that at least 2 Gig of RAM is needed and preferably a dual core processor. You may be fighting an uphill battle, although I noticed a few posters on HardOCP that have similar boxes as yours, with X1950 Pro cards, who claim they get playable frame rates at mid to low settings.
A previous post mentioned overheating which will definitely cause artifacts. Do you have good case ventilation with good flow-through? What brand of 1950 do you have? Tmoney468 indicates that he had problems with a Sapphire branded 1950. After reviewing numerous web sites, I came to the conclusion that the VisionTek (one of a few with a lifetime warranty) and GeCube (they look the same and have caps intead of VRMs) seem to be more reliable but do not have temperature sensors or a way to adjust the GPU fan speed and only come in the 256 Meg variety.
If you run the card in another box, and if it craters there, then return it.