The minimum recommendation is always way higher than what the actual card requires, because when making that recommendation they have to include the rest of the system, and they always guess high.
If you look at the actual power use by just the card the GTX680 only uses about 186w, or ~15.5 amps. The HD7970 GHz Ed. on the other hand uses about 238w, which is ~19.8 amps.
My guess the problem is that the HD7970 is hitting the over current limit on the 12v rail and the PSU is shutting the computer down, since a single 12v rail is limited to 18A. My guess is also that both PCI-E power connectors are on the same 12v rail. A poor design, but not exactly uncommon in cheaper power supplies. It isn't uncommon to see the CPU and Molex/SATA connectors on one rail and the PCI-E power connectors on their own rail. Or even just the CPU on one rail and the PCI-E and Molex/SATA on the other. In this case the GPU would be hitting the power limit and shutting down the computer.
Either way, I'd look for a power supply that is better, as that Seasonic barely qualifies as a 500w unit. In fact I've seen 500w units with more balls than that Seasonic.
So yes, if you are looking at a TX750, go for it. That is an extremely good PSU, and way better than what you have now. And if you can, overclock your CPU. The Q6600 is going to hold you back, but if you push it to 3.0GHz+(and they easily do that) then the bottleneck will be a lot less noticeable.