A shot in the dark, I would say that your Power supply is your culprit. It cannot provide enough power when you are pulling power for your games.
I am guessing your Power supply is also around 5-6 years old. It is barely adequate for your computer, even if new. Now it is aged, and could be providing
less performance than what you need.
If it cannot provide proper power, it can cause VDrop or VDroop on your motherboard as your processor pulls more power, causing lower voltage. This can shut your computer down.
It can also damage other components in your computer.