If the engine fan does not run when the A/C is on, the A/C cant rid it's condenser of heat, therefore, it cant take heat out of the air in the cabin, so that makes sense.
When you MOVE the car, air flows over the condsenser anyway so of course the air out of the vents turns cold. No mystery there. The mystery is: "Why won't the damn fan run?"

The ECU/FCM relationship is simple in concept, but sometimes we see failures that make no sense.

Here is how it all works:
The ECU sends "tickle pulses" to the FCM. The FCM responds with an "I'm here" pulse. If the ECU wants the fan to spin, it widens its tickle pulse in proportion to how fast it wants the fan to run. The FCM sends a pulse back to say "Im not only here, but I will obey".

• If the ECU does not get an acknowledgement pulse back, it knows the FCM is in trouble or broken - so it lights the little radiator light on the left side of the instrument panel.
• If the FCM sends a modified (Wider) pulse back, the ECU knows this means the FCM is having issues or sees improper current (too high or too low) going to the fan motor - so it lights the little radiator light on the left side of the instrument panel.
• If the FCM does not get a tickle pulse from the ECU, it enters "Emergency mode" and spins the fan at warp drive speed to protect the engine from overheating.

Thats how it works. ANytime you see that light, the owner's manual tells you to take the car to a dealer and write them a blank check. Make sure you NEVER do that.

There are six wires going to the FCM.
Two huge red/black battery feeds, the red is fused in the little black box next to the battery at 50 amps and is labeled for the Fan.
Two huge red/black feeds to the fan motor.
One little wire called "Ign" that supplies "wake the hell up" power from the actual contacts on the ignition switch.
One little wire called "pwm" that is the bi-directional signal wire to the ECU.

Something I just learned after all these years: If you cut the pwm wire and ground the end going to the FCM, the fan should run. This is a built in diagnostic in the FCM. DO NOT ground the end going to the ECU.

First question for you: Is your little radiator light on? If it is not on, cut the pwm wire, if the light still does not come on the ECU is bad or the light is burned out. If the light does come on, this means the ECU seems to be OK and your FCM is bad.

THe big question is: Is the 50amp fuse good? (Must be if you jumpered around the FCM and the motor ran.) So, is the little radiator light on or coming on at times? If so I'd blame the FCM, if not I"d lean towards the ECU but ECU failures are super rare.