Do I need a new PSU?

[o_O]

Check that your fans aren't too dusty. This happens quite a bit especially with laptops.

As for the test, do an experiment. Run a high-end 3rd game at full quality and see when it turns off. Then restart the PC and leave the empty desktop open for a bit. If it goes off on the first but not the later, it's the overheat.

The more demanding the task, the more power your components will draw from the PSU, so this test won't work.

I think it would actually. The difference between the power demand and the overheat is quite simple really. Time. Once you start a high-end application, the power will be immediately needed in all of the components, save the cooling system, and thus you'll see the effect right away. If it is the overheating, you'd still have to wait for a couple of minutes for the hardware to heat up to the level where it starts overheating. Either way, you should see that it's not the PSU once all the fans are working on full power and the PC is still on.

The reason I said it wouldn't work is because spontaneous shutdowns aren't a usual symptom of a faulty PSU. You're much more likely to see lockups and hear funny noises from the optical and hard drives. That's because the highest-drain devices on a PSU are the ones which require power for kinetic motion.

If it's overheating, unless you have an old CPU, you're going to see random shutdowns because pretty much every PSU since 2001 has had automatic shutdown capabilities beyond a certain temperature, to save it burning out.

Given the number of devices on Omecle's PSU, only a faulty one would be weak enough to not be able to power them.