Engineoverheats but still blows cold air - Air Conditioning and Heat for GMC Terrain

Your statement in it-selves solve the problem.
Your vehicle cooling system and air-conditioner system work independently.
Your vehicle cooling system is driven by the v-belt that turns the water-pump and circulates the coolant within the engine through the radiator and back into the engine and flow is controlled by a thermostat.

Your vehicle air-conditioning system operates from a v-belt that drives an air-conditioning pump which circulates coolant through a sealed system and generates cold air completely independently from the engine cooling system.

It seems that you need to repair the engine cooling system which has a fault and this fault could be
a) combustion seeping into cooling system (blown gasket),
b) cooling fan not operating (this fan engages at specific engine temperatures and suck cold air through the radiator in order to cool the engine coolant as it passes through radiator veins),
c) faulty cooling thermostat (thermostats work mechanically and through temperature expands and opens the water circulatory channel / opening, and in cold weather restricts this opening in order for coolant to allow engine to reach operating temperature and then allows the coolant to circulate freely),
d) engine combustion could be too hot due to fuel mixture discrepancies (too lean) thus overheating the internal combustion process which will overheat coolant,
e) engine cooling system leak causing the system not to seal thus restricting the pressurised coolant circulatory system (the system needs to be pressurised for effective circulation as well as raising the boiling point of coolant (under pressure water would boil at 110 degrees Celcius instead of 90 degrees Celcius).

Hope you find the problem soon as overheating is detremental to the engine and all integrate parts associated therewith.