I need PC upgrade help

[Nimon9628]

Hi guys, I really want to be able to game on my pc but its just too old and I'm not sure where to start in terms of upgrading it. 
My budget is no bigger than $350 but I'm willing to go under if possible. 

My current specs are: 
Processor: AMD A10-5800K APU
Graphics Card: AMD Radeon HD 7660D
Memory: Corsair XMS3 DDR3 2x4GB 
Mother Board: GIGABYTE F2A75M-D3H

[Issues] 
At the moment, whenever I try and play a game my entire computer shuts off within a few minutes. I've looked up temporary fixes online and lowered my maximum processor state (under advanced power options) which worked for a while but no longer does.  Some USB devices occasionally don't work as well so if the best option for me is to replace the Mother Board then I don't mind. 

I would really appreciate some help as I am very new to all of this stuff. Thank you.

[Phentos]

It's time to retire that PC completely.

 

You sure you can't squeeze some more money into your budget? $350 doesn't get you much in today's PC market.

[JoostinOnline]

Nothing you can do about upgrading that machine. My suggestion is to buy a used Optiplex with an i7-2600 (make sure it's the regular width ones) and 4GB of RAM for around $150 on eBay, put a new power supply in it that's at least Tier 3 (see the link in my signature), and look for a used GTX 760 or 950 locally. Put your 8GB of RAM and hard drive in the Optiplex. Sell the 4GB of RAM that came with it. You'll have a much better computer under budget. Spend a little extra on the graphics card if you want. 

 

This is a good option:

https://rover.ebay.com/rover/0/0/0?mpre=https%3A%2F%2Fwww.ebay.com%2Fulk%2Fitm%2F142767801766

 

Edit: For the record, I'm almost 100% sure this is a failing power supply.