A 7970 would be a good choice. I run a 7950 in a 3+1 configuration. Yes you can continue to use the 4th monitor while gaming in EyeFinity. The extra ram on the ATI cards vs the Nvidias does help with multi-monitors. When running EyeFinity resolutions I've seen my card go past 2 gigs of ram useage. While the Nvidia 670/680 are good cards, at higher resolutions (multi-monitor) the 7970 tends to be a better choice. The 7970 should fit in your case just fine and your 750 watt PSU is more than enough.
Ok which 7970 would be the best the 7970 ghz edition or should I opt for the 6gb 7970 version, if so does the increased performance justify the price of the 6gb version and will I ever use the 6gb of vram or will the card bottleneck before it gets to that point?
While the ghz edition is nice, any 7970 with a descent 3rd party cooler should be able to overclock and hit the same levels. I would recommend a Sapphire or Gigabyte 7970. Both have good fans and overclock well. My Sapphire 7950 easily overclocks past a stock ghz 7970.
As far as the version with 6 gigs of ram, to me it seems like overkill right now unless its the 7990 (i believe thats the one) which is a dual card on one board. (3 gigs of ram per card). Right now during gaming on one screen most games stay under 2 gigs of ram useage (there are exceptions like Skyrim with the HD pack and add ons). During EyeFinity gaming I have seen a few games go past the 2 gigs of ram useage but usually not by much. The highest i've seen was using 2.8 gigs of ram running Alan Wake in EyeFinity. I dont really see 6 gigs of ram being an advantage in a single card solution. By the time you got to where you needed that much Vram, it would most likely be time for another card.
Someone mentioned crossfiring your 6950. This would actually give you more "raw" power than even a 7970, the only issue would be the Vram. The higher the resolution the more Vram you want especially when you go up to an EyeFinity resolutions (5760x1080 for me). Personally I would take a single 7970 with more Vram over Crossfired 6950's even though they may give you more raw power.