what termostat does is switching contours for the coolant, when engine is cold coolant cycles just on the short countour and when engine is warmed up it switches to the regular contour, if it would be stuck in cold position the engine would overhear while if it would be stuck in the warm position engine would take much more time to warm up. Thermostat does not infuence temperature inside the cabin in any way. you prolly thinkgin about the heater core not the thermostat.
i mean how much colder engine should get during the stops so you would acutally feel it? you would see that drop on the temp gauge then. i had the same problem in my car
when during stops the openings on the dash would blow only cold air and the ones on the bottom still would blow hot air, the first thing to trouble shoot is the temp sensor inside the cabin, just put the temp to max hot and check if the problem would persist, if it will then check if cold air comes out from all the opening and not just the front one on the dash. openings next to ur feet are of a particular importance, compare the temp by hand with the dash openings. let me know o the results.