Its possible that the car is actually getting up to temperature then. Has your shop checked the actual coolant temperature after the car has been running a while? And did they check the coolant temperature sensor? Someone needs to do those things to find out if the coolant is getting hot or not. Someone should also find out whether or not the heater hoses are getting hot - both sides - to see if the heater core is actually flowing.
Thermostats can be had in different temperature ranges. A typical OEM thermostat would be a 180-200 degree or so. But one can get lower temp thermostats. Was the new thermostat OEM?
Maybe you have 2 completely different problems - a bad coolant temp sensor and a stuck blend door in your HVAC ducts. This is really a stretch - especially b/c what you describe sounds just like a stuck open thermostat - but if the thermostat has been checked and double checked then there are other things to look at.