Overheating when driven but can idle and not overheat

What is your question?

2000 LHS 3.5L will begin to overheat while being driven, but then put vehicle in park and it can sit and idle a normal temperature indefinitely. There is a very faint coolant smell, no visible leaks and the cooling system is free of air. Engine runs great, even when temp gauge says it's getting hotter. The engine has 135,000 miles on it and the vehicle is in superb condition. Maintenance records also indicate the car has been well taken care of.

What makes this problem better or worse?

Turning on the heater, putting car in park and letting the engine idle

How long has your 2000 Chrysler LHS had this problem?

I just bought the car, but as I understand it rhe previous owner had a similar issue. But after replacing battery and thermostat, the car wouldn't start so he donated it.