it supports that resolution. Think along the lines of it dying a slow death.
it always sad when a good GPU starts fading away. My Old Reliable X800 is still strong although.
Maybe check your PSU. The X800 still draws a bit of power. I dotn know the exact values atm, but, from what you are saying a pretty good diagnose is when the CPU/GPU needs the extra wattage the PSu fails to deliver/level the amps need and just shutdowns itself protecting itself from the sudden "peak" and the rest of the PC with it.
I guess you have a bad PSU !!! Don't say you have a bad X800 pl0x !!!!
PS: I'm getting old and sentimental. Blah...