I have a Precision 7510 running Linux and don't have any of these issues so I wonder how different those two series actually are.
The 5560 is between 7.7mm (0.30") (front) and 11.64mm (0.56") (rear) thick. The thickness of the 7560 is between 25 mm (0.98") and 27.36 mm (1.08"). The increased thickness should solve (or help solve) the overheating.