PC shutting down due to CPU overheating

Yes, because before thinking about buying anything, you need to check that first.
Here's a method just in case:
You disassemble your cooling fan to access the processor. Then you remove the processor from the motherboard.
Using a cotton swab and 90°C alcohol, clean the surface of the processor thoroughly until it's as clean as possible. Do the same with the underside of the cooler that makes contact with the processor.
Once done, make a small ball of new thermal paste in the center of the processor. Then put it back on the motherboard, reinstall the cooling fan (trying to properly position it directly; if you move it once it's in place, it may spread the paste in a weird way).

You don't necessarily have to remove the processor to do all this, but I personally find it more convenient.

And afterward, take the opportunity to thoroughly dust the PC, inside the fins of the cooling fan as well as the blades of the various fans.

In any case, don't worry, overheating shouldn't impact the condition of your components, at least not unless it’s for a very prolonged period; you can't just ruin them like that, there are all the necessary safety measures in place to make that impossible (the processor shuts down without asking your opinion before reaching a temperature that is really dangerous for it).