I'm wondering if this is the computer overheating, or if it is my power supply failing.
The simple answer is "yes", one of those two is the most likely culprit.
Download CPUID's Hardware Monitor (to report core and gpu temps), Furmark (to stress the gpu), and Prime95 (to stress the cpu, and test memory issues as well).
Start with Furmark, watching gpu temps and see if they become problematic. Then Prime95 (make sure "detect rounding errors" is checked), watching core temps.
If gpu or core temps get too high, you've identified the culprit. If Prime95 fails due to an error, rerum it with one stick of memory at a time to try and find a bad memory stick.
Let us know what you find.