Sounds like your GPU is crashing. Unfortunately the HD6xxx series has reached peak optimisation and been moved to legacy status, there will not be any further driver updates for your product. More information available here.
The latest Crimson driver for your system is 16.2.1 and is available here.
The last Catalyst driver for your system is 15.11.1 Beta and is available here.
I would recommend using DDU to sweep your system of AMD drivers, and then trying the drivers above to see if they resolve your issue.
Sounds good. I will try the above steps if I get the error again. Oddly,
after posting my question, I let Windows attempt an automated fix, and I
haven't had the issue reoccur. We'll see if it actually worked.
Unfortunately, I didn't catch the repair performed. Thanks again for your