I would say your GPU heatsink fan is out. When it does that 10 minute to 10 second shutdown decay, does it last longer if you stop gaming/using the computer? Try this out and see if it goes back to even 5 minutes.
If it does then most likely your GPU fan just died and cant cool off your GPU, although some low end cards dont even have fans on their heatsinks so if your GPU doesnt have a fan then I'd say your CPU heatsink fan is dead.