The GPU is better for gaming, so I would say to dump more of your money into the GPU instead.
You should get the FX-6300 and a good CPU cooler, to overclock the 6300.
Then, with the money saved go with a stronger GPU. It will be better in the long run.
If I could ask, what is your budget for this upgrade?
Oh, and if you go against my recommendation would prefer to get the 4440, then I would recommend getting the 4570 instead. It has a small price difference between the two and is a better processor.