A 480 8GB is viable and is a decent upgrade on it's own, but that card won't stand well in the future with your dated Phenom II even if it is a six-core. Hopefully you're not playing Bethesda games because they run like crap on cores that aren't all about single-core performance.
(I made the decision to go from Phenom II X6 @ 4GHz to I5 2400 the FPS boost was huge, though still only 60% usage at max and 80-90% with a powerful mid-range card in today's titles, it was a slight upgrade towards multi-core performance, but a big one for single-core. Anything higher than a GTX 970 will start bottlenecking it on some games, only an upgrade to an i7 or an unlocked i5 would save me from seeing the bottleneck at all, but I digress.)
Otherwise you're good for another 2-3 years tbh, since we still aren't seeing the full benefits of DX12 (sometimes makes games run worse) and even latest additions to DX11 in newer Windows releases (because DX11 isn't getting replaced by DX12, they go side by side).
I think more than 4 cores is only a good thing in your case when you have a Phenom even if it's probably not even clocked as far as it can go. But as I said, it's still pretty good since Phenom's are slightly better than the FX line of CPU's in terms of performance.
If you can't get that raw performance at least you got enough cores to make up for what it is. GPU's for gaming is number one in priority in terms to increase performance in most titles, so yeah 480 8GB ftw.
The 7970 still holds it's own as well, hence I voted to keep it until you're sure you want around 30-40% of increase in raw performance give or take.