I have a 2000 DODGE Ram 1500 v8. My heat only work while its blowing low and when I switch speeds it starts to get cold. I almost checked everything that could be check other than the wires.
I'm pretty sure the thermostat is stuck open. The coolant/engine won't heat up fully and you'll get cool coolant through your heater core. In most vehicles (and I'm pretty sure this is how the Dodge Ram's work) the coolant goes from the engine to the heater core directly. If the engine is running cold due to an open thermostat, only cooler coolant will be making it to the core. Your description of the heater putting out warm air when sitting still, then going cool after rolling fits right into this scenario.
Is it a gradual change? Like when you turn the speed selector dial, does it slowly get cold? Or does it just literally instantly change? Are there any noises you hear when you change from low to different speeds?
This is important. I believe they use HVAC control units in those and that may be causing the problem. There are two motors, one for vent control and the other for your blend door. If there is an issue with the control unit, it can act up. I've seen similar issues electrically in many Dodge vehicles. Dodge (Chrysler) is known for having issues with their HVAC (Heating, Ventilation, Air Conditioning) systems.
Some cars have a small electric pump for creating coolant flow around the heater matrix. I don't know if your car has one of these, but if it has, it could be the problem.
I have a 1999 dodge ram did the same thing. It was the heater core. Mine would blow heat on low but no other speed. Once I replaced the heater core everything worked great.