I have a very-recently-acquired 94 Concorde that is doing the exact same thing, except that it does it right away, long before full warm up. Coincidentally, the HVAC blower will will not work while the controls appear to be working properly. BCM perhaps?
The 1st gen ('93-'97) LH fuel pump and sender assy was easy to access by removing a plate on the trunk floor. You don't have to drop the tank. A test would be to briefly ground the sender wire and the dash gauge should move towards full. This tests the circuit from the sender to dash.
The DRB II/III scan tool is best for ATC/BCM diagnosis. Generally the BCM is OK and the problem is in the ATC head itself. Display lighting can be caused by broken solder joints or failed internal VFD power supply.