Especially as you say fallout 4 which needs a far more powerful computer to run, runs better than this. (drivers, 3rd party background program, corrupt mod etc). Hardware wise you should be fine, got to be an issue with your software somewhere.
Pesonally though I have no idea why your getting such slow speeds. With a 960 on ultra settigns, with fraps still getting basically 60 (yes it does drop to the low 50's occasionally however remember you can loose over 10fps while recording. Check mortherboard support site for drivers and bios, its not the first time we have seen AMD user complain over. Without fraps getting 50-70fps, with a much much slower gpu. (even the single core performance of the AMD while poor compared to today's intels is much faster than the intels back then) The recommened cpu back then was just a dual core e5200 2.5ghz which the 6300 leaves standing in its dust.
This is an old game, of couse an fx- 6300 should be able to max this game out 1080, its a six core 3.5ghz cpu.
