Currently Have
CPU: AMD Phenom II x4 965
Mobo: Gigabyte GA-970A-D3
I originally built this PC in May 2012 and it's been going strong ever since. Last December I started having random crashes when I would run a game/video that also had audio playing. After talking to numerous people we determined it was something to do with the audio chipset; I sent the board into Gigabyte and they replaced the physical chipset as well as updated the BIOS, and this has seemed to fix my problem for the most part, but I have had a few crashes since then very similar to those before.
I'm currently looking to upgrade to either an AMD FX 6300 so I can keep my current mobo (cheaper option) or upgrade both to hopefully avoid any future crashes all together, and am considering an i5-3470 and an ASRock H77M Micro ATX mobo. I will not be OCing the CPU and will never have dual GPUs so I don't necessarily need a Z77 chipset mobo.
I am leaning towards a new CPU and mobo entirely just to more "future-proof" myself, but I appreciate all arguments for or against the change.
Relevant links (parts in consideration)
FX 6300 - http://www.newegg.com/Product/Product.a ... 6819113286
i5-3470 - http://www.newegg.com/Product/Product.a ... 6819115234
ASRock H77M - http://www.newegg.com/Product/Product.a ... 6813157303
Other parts (for w/e reason)
GPU: Sapphire Radeon HD 7870 GHz Edition (model: 100354OC-2L)
PSU: Corsair TX 650W 80+ Bronze (model: CMPSU-650W)
Ram: Patriot Gamer 2 series 2x4GB DDR3 1600
I want to keep this under $250 as well (or w/e the i5 + mobo cost is depending on mobo price).

Edit: I would also consider maybe an i3-3220 + mobo or i3-4130 + mobo as well. I don't necessarily need the power of the i5, as the most intensive thing I will do is gaming and even then I don't need to be running at 60+ fps all the time; being in the 30-40fps is perfectly fine with me.
Thanks in advance for any advice! o/