Some updates in case anyone with the same issues come across this thread:
I tried the computer with my TV using a HDMI interface to rule out any problems with the card. It worked completely fine on it, which leads me to believe the problem is the VGA to DVI adapter, or just the analog VGA cable 'incompatible' with such newer cards. I will get a DVI - DVI cable tomorrow (monitor doesn't support HDMI) to see if it stops.
I'm guessing the move to DVI will help. The screws on all of the connections are tight, right? That could be the cause too.
Yeah I've double checked all that. Another thing to note is the flickering doesn't show up in screens or recordings (fraps). Does this just confirm is a problem with the monitor(/cable) and not the computer itself?
Yes. If you don't see the issue in a screenshot/recording, then the GPU is rendering properly (and the vram is storing the scene properly), and the issue is introduced as it gets shipped to the monitor.
It's possible the GPU has a problem in it's output controller, but it's much much more likely the cable or monitor.