The A8-5600K is certainly bottlenecking your GTX 660, and would bottleneck a GTX 770 even worse.
Seriously, replace the CPU. I used to have a Phenom II 955, which is actually about the same strength as the A8-5600K, and it bottlenecked even my GTX 460 in some games (that was my old build).
Later I switched to a GTX 660 and kept my Phenom II 955. Horrible mistake. My framerates barely improved. Replacing my Phenom II 955 (which, remember, is about the same strength as the A8-5600K) with the much stronger i3-4360 moved my Dragon Age Origins framerate from 45 (minimum) to 200 (minimum), just as an example.
My brother until very recently had a Phenom II 1090T, which is stronger than the A8-5600K, and it bottlenecked his GTX 560 severely. His framerates doubled in some games from switching to an i5-4590, even while still using the GTX 560.
You ofc can't expect your framerates to double in *every* game, as many don't use the CPU that heavily. But you're certainly getting a CPU bottleneck right now, so replacing that CPU would help to some extent in many games, and without replacing the CPU it would be pointless to switch to a GTX 770.