Ok so here's where I'm at in terms of testing:
1. Swapped out PSU with my sister's 700w OCZ PSU. The computer boots up (I can hear the normal sounds of booting to the desktop and I can see the home streaming popup from Steam on my laptop) but all my monitors are blank.
2. Swapped back to my normal PSU. The exact same thing as 1. happens.
3. With my normal PSU, pull my GPU out and plug one monitor (DVI) into my motherboard. Everything seems to work fine (it boots and I can see my desktop, launch TF2, etc.).
4. Install my GPU into my dad's PC. Everything seems to work fine.
5. Reinstall GPU into my PC, using a different PCI-E lane. Only one monitor (the same one as 3.) plugged in. Same problem as 1.
6. Swap out GPU power cables from my PSU to the other one that came with the PSU that I have never used (card is 1x 8 pin). Same problem as 1.
So I'm kind of stumped. It seems that the GPU works (worked in my dad's PC), the PSU works (worked when using integrated graphics, also tested with a different PSU and had the same problem), and the monitor works (worked when using integrated graphics), but all together something is messing up. The only thing that I could think of is that all of the PCI-E lanes on my motherboard aren't functioning correctly?
EDIT: Resetting CMOS with my normal PSU plugged in let me boot into Windows and have my monitors display things fine. But the problem in the original post (choppiness) persists. It is worth noting that resetting the CMOS and trying my sister's 700W PSU does not result in my monitors displaying anything, so I guess that PSU isn't working correctly (she hasn't used it in like a year). I'm going to pull my dad's PSU out of his rig and see if my PC starts acting normal again.