There are three temperature sensors or switches on the head and/or manifold that I know of. Starting from the front of the car and working backward they are: Temperature switch, Coolant temperature gauge sensor, and Coolant temperature sensor.

I don't know which if any of these are relevent to my issue and if so how to test them. My guess is the Coolant temperature sensor which has as I recall a four pole connector on it. I suppose I could take swap the one in my 300 TE and put it in the 300 E but I'd rather just test them if I knew how.

I suspect the sensors are way less money than the cold start injector.