I suspect your engine overheating is not part of the rear gears issue. I am pretty sure you have to hit the button to go any higher up the gear selector past drive. I do not remember every being able to dump it up into N without the button and really why would you.
I would get the car to operating temp at idle, watch the cooling fan and make sure it goes on with the car in D brakes on just sitting.
I would check reservoir and radiator for leaks and check the lvl. I would also while cold check to see if fluid is circulating in the radiator to ensure the water pump is pumping.
Other than that other than a thermostat or (very rare for 30K+ miles) head gasket. I cannot imagine it is anything else.