Alway hot air not cool 2003 chrysler

1. Engine Coolant Temperature.
If the thermostat in the cooling system is getting stuck in the open or partially open position rather than opening and closing as needed to keep the coolant at the required operating temperature then you will get the results you have described here with the coolant temp dropping as you drive. The radiator is providing optimum cooling when the car is moving and there is plenty of cold air flow through it.
I am assuming the temp gauge in the vehicle is reading correctly.

Thermostats in the cooling system do become faulty in time and it may well never have been changed (or only changed once) since the car was built and is possibly gummed up with debris. With those symptoms I would definitely change the thermostat. Make sure a full flush of the cooling system is done at the same time and refilled with fresh coolant.

2. In Car Heater.
I would not worry about the interior heating until you change the thermostat on the cooling system. However if you continue to get cold air from the heater then there will be an issue with hot coolant from the cooling system being restricted in flowing through the heater core.

Your actuator is bad on the vent door, that switches from cool to hot. some units have several of these, some only have one

Need to check the blend door actuator...not easy job to check it out as well as it is costly repair.

If the cooling system is functional, yes.
The ac condenser adds heat to the air coming thru the radiator when the ac is on. If the engine cooling system is weak, the extra heat will make the engine run hot.
Low coolant flow thru the radiator is a common cause.

When the coolant flush was done, did you have the thermostat changed as well? I would have the thermostat replaced. If the temperature gauge was reading properly, but you still had warm, not hot air, it would lean more towards the heater core being clogged, but that deosn't seem to be the issue your having, so i would go with the thermostat replacement. And whereever you have the repairs done at...make sure they fully bleed all the air out of the cooling system, this could also cause poor warm-up / heating problems. Hope this helps...

Is your A/C off? Make sure it is and try it-good luck.

there might air in the system so you will have to bleed it or remove the rad cover and let the car run for a bit the should be on in full

My car has this resistor mounted under hood in the AC plenum housing....mounted by two screws. Has a wiring plug. The resistor elements are large resistance wire coils. Visual inspection will show burned coil. Trying to solder these (If open) will not work....they run very hot and are cooled by placing them in the fresh air intake of the plenum. Replacement is best....parts house may have them. If not, try Junque Yard.

sounds like a restriction in the coolant bypass and or heater core. you will want to check if both sides of the heater core hose near the firewall are hot on both sides if so you have an issue with the air ducts but if not then you may have a failing heater control valve or a restriction.

CHECK COOLANT LEVEL. COULD BE LOW. DONT OPEN WITH HOT ENGINE, WAIT TILL ITS COOL DOWN.