I noticed your monitor is 1920x1080p I believe and your power supply is 750 watts. So my post is directly to your set up.
Here's some things to think about. A 6970 has a 6/8 pin whereas a 6950 has a 6/6 pin connector. A 6970 is more capable of higher voltage and over clock than a 6950. A 6950 cannot reach the performance of an over clocked 6970 regardless of what some might say. If you do unlock a 6950 make sure it's the 2GB version. The 1GB 6950 do not. Flashing the BIOS also voids the warranty if the card burns out before you can get it back to original BIOS. If it burns out you can't get it too late to revert it. IF you want a 6970 performance I'd get that over the 6950.
Read this thread on OCNThe 6950 to 6970 unlock thread
and this one off ITShoutOut.com Turn your Radeon 6950 into a Radeon 6970
to familiarize yourself before you go this route just to have hind sight of what to expect.
Now for the other side of the coin. The AMD 6970 is a bit better than a GTX 570 and much more comparable in performance. The GTX 570 is about 15% less performance than a GTX 580 at about 70% of the cost. Also a good choice for price per performance. I'd concur that the GTX 580 is a bit over priced, but if you want the best single card period the GTX 580 is the card for single card solution. However with your current CPU it may be too much and you will bottle neck it.
If your going to upgrade the CPU eventually than the 6970 or 570 will be a good choice in my own opinion.
A lot to think about but you should choose what you feel best suits your needs or what suits your budget to your needs. Which ever is more important to you. Read the threads and reviews online to get some insight first before making up your own mind.
NVIDIA GeForce GTX 580 Video Card Performance
AMD Radeon HD 6970 Video Card
NVIDIA GeForce GTX 570 Video Card Performance
PowerColor AX6950 PCS++ Video Card
NVIDIA's GeForce GTX 560 Ti
Good luck and let us know what you ended up doing. Always curious on the outcome. All the cards are good cards and can play todays games without any problem.