I'm trying to diagnose an issue I'm having where my computer shuts off during certain games. My first instinct is that the CPU is overheating. Running 'sensors' show that this may be the case:

And this is after browsing the web -- no gaming. Should I take this at face value and invest in better cooling for my CPU? I read an article once that said the values sensors gives aren't entirely accurate for AMD products so I'm not sure what to think.

It happens consistently in the following games:
Pillars of Eternity
Borderlands 2
The Elder Scrolls IV: Oblivion (wine)

Other games in which it has never happened, even after hours of gameplay:
Team Fortress 2
The Talos Principle
SOMA
Torchlight 1 & 2
Minecraft

I just find it odd that this happens only in certain games and not others.

I need help to diagnose this so I can fix it once and for all. The CPU is an AMD FX-6300.

Sometimes cleaning out the dust is all you need to do. Replacing the thermal compound on CPU would be the next.

First of all, think of it this way... If the sensor is failing and showing higher results, this means that the shutoff is done earlier..

On the other hand all of what you describe are symptoms that might just happen when thermal paste gone bad and in need of a replacement... I would trust the sensors here, since they fit ... But, ofcourse, the cooler might be failing, or even worse things might have happened that would also fit your symptoms... But those problems are difficult to diagnose on a forum...

You should also keep in mind that some games might be more CPU intensive than others even if it wouldn't seem that way on the first look (depending on how they use the video card and what your video card supports... It's not uncommon that unsupported functions from your graphics card to fall-back to software rendering processes that overload your CPU and overheat it)... Then there is a matter of multi-threading support, etc... So don't be suspicious by the fact that some games heat your processor and some don't.... If anything, this should just reinforce your believe that the heating is the problem here

I did replace the thermal paste a few weeks ago. This computer is relatively new and still quite clean inside. It has also shut off while doing nothing but browsing the web, watching something on youtube or twitch.

Where it says "crit" in the output, is that the temp at which the computer shuts itself off?

When the system boots, you might have a screen that shows your CPU temperature, or perhaps that data is accessible in your BIOS. Comparing the values there with "sensors" could give you a clue.

Obviously this doesn't prevent the actual sensors (the probes, not the sensors program) from reporting wrong temperatures, or the BIOS from wrongly processing the raw temperature data.