Hi Elessar, like you I am a big time follower of this page for many years, but haven't really responded to or posted many things here. I found myslef in a very similar position to you not long ago so I thought I would share my personal experience and upgrade path.
I had a Phenom II, a Sabertooth motherboard and an older GPU. Now you have a newer nice GPU, so you don't have to upgrade yours like I did, but if you have a good motherboard I would suggest upgrading to the FX 8370, that is what I did and I love it. I bought mine at Micro Center for well under $200 and it would be the only upgrade you would have to do (maybe a bios update for you MB).
I know there is going to be many a person suggesting to go to Intel, but for your budget your going to be getting bottom of the barrel cheap MB and probably an older i5 or an i3, if your lucky. With a bottom of the barrel Intel MB your not going to get a very good overclock and may cut the overall performance of you system. I've seen guys with high end i7 rigs that cheaped out on power supply, mother board, ram, and GPU and wonder why I'm getting better gaming experience on my rig (being AMD). Yes, Intel will out game AMD on most older games but only if its got the supporting hardware to do so. By the way every game that I've played since Witcher 3 has my rig and my best friend (who has an i5 Haswell rig and 980 GPU) at nearly the same FPS. I play Crysis 3 at the highest settings with my FX 8370 and Sapphire R9 290 overclocked (the rough equal to the GTX 960/970) never had a stuttering problem in game. I also max out Witcher 3 and Total War Attila. Not really playing much other games right now to tell you how they do, but I'm quite confident with my rig.
One last thing, a lot of people will say there is no difference in gaming between the FX 6300, FX 8350 and FX 8370, but looking towards the future that won't be the case. DX12 games will allow all 8 cores of an FX to communicate with your GPU so FX 8350 will outperform the FX 6300 in future games. The FX 8350 is a lower binned processor than the FX 8370 so you can overclock FX 8370 to the performance levels of the 9590 much easier at lower voltage, lower heat. The FX 8370 has also been tweaked so that it uses less voltage than the 8350 (at stock and overclocked) and has slightly better performance. With your budget at $200 and not needing a new GPU I would jump on the FX 8370. Realistically an FX 8370 and GTX 960 4GB will keep you gaming through the lifespan of the Xbox One and PS4 as games are typically "held back" so that console systems can play them (more console gamers than PC gamers). A gaming rig with FX 8370 and GTX 960 4GB is vastly more powerful than a PS4.