It sounds like you've tried switching which monitor is plugged into what connection on the graphics card, and when you did, the corrupt image moved to the other screen.
This would make it clearly an issue with the output from the computer.
If your screen shot that was taken with the display output in a corrupted state is accurate when viewed on functioning equipment, this tells us that your GPU memory isn't actually the culprit in this case, which is the first thing that comes to mind when I see odd behavior like that. That leaves the GPU chip and the output stages on the card, or the systems powering them.
I would grab something like MSI Afterburner and down clock the GPU, reducing it's speed in 100 MHz increments and see if the...