One possible cause of shutdowns is that the power supply or the CPU is getting overheated.
Assuming the power-supply fans, the video-card fan, and the CPU fan do work, next try installing Asus Probe and opening it up. What's the idle CPU temperature? Leave Asus Probe open, fire up WOW, and play a bit, while using the ALT TAB key combination to switch back to Asus Probe. What's the CPU temperature reaching after a few minutes of gameplay? If the CPU overheats, it can shut the system down, so hopefully your CPU is staying under 60C. A cool heatsink can simply mean that the heat isn't making it out of the CPU into the heatsink, so don't judge just by the heatsink itself.
If/when the system shuts down, also feel the power-supply unit. Is it hot? Power supplies max out at less wattage when they get hot, and are generally able to shut themselves down when they overheat & overload, so if the combined heat output of your CPU and your video card are getting the power supply too hot, it could shut down for that reason. Improving case ventilation may help, but your system shouldn't be pushing a true 600-watt PSU to the brink.
Power-supply quality can vary widely. Sometimes they're rated way too optomistically on their wattage. Sometimes they just aren't stable. If it's a cheap or freebie-with-case power supply, and you can't get this resolved, maybe buy a recognized high-quality brand like a Seasonic or Fortron.