I think your build can handle the FX 9370. Being at 4.4 GHz Base at 1.53750V like the 4.7 GHz FX 9590, your CPU should draw slightly lower power than the 220W of the 9590.
You should be more concerned about your CPU temperatures to be below 61.1°C, which is possible with that AIO of yours.
You should also check your VRMs on your motherboard if they have heatsinks. If they do not have heatsinks, I would be surprised your board could survive with that CPU. But just Ziptie some little 40mm or 80mm fans from some $10 laptop coolers and you will be fine if you dont overvolt it further.
You might even reach 4.7 GHz easily with 1.53750V, heck mine reached there at 1.49375V and is now cruising at 4.9 GHz at comfortable temperatures at 1.60625V. If temperatures are already high, you might want to disable 4.7 GHz Turbo Core (only works when 4 or fewer cores are utilised) and undervolt to around 1.4-1.45V or lower at 4.4 GHz.
Watch the temperatures, I need to stress this as it is the main cause of lockups even with sufficient voltage.
Your PSU is sufficient, provided that your CPU is not sharing a 12V rail with the motherboard or GPU (if using multi rail PSU). If yours is single rail you are fine with your current build. I recommend 650W or more for this, 750W if you plan on overclocking, 1000W if you plan to crossfire another RX 480 in the future and overclock. Plug in any extra "for extreme overclocking" CPU power wires as these are necessary to supply the increased power draw of the FX 9370.
It is quite amusing how so many people try to dismiss the problem as impossible and coax the OP to sell his build and go to a Z170 platform. Problem is, most people dont want to spend the unnecessary costs and time. It may be easier to say buy X buy Y, but isnt itbetter to reduce waste and try to make the most of what you currently have?