Well I have a 1996 crysler cirrus lxi and well I had the car running and the electric locks would not lock the doors while it was running, it would unlock the doors but not unlock. But also when the car is off they work and itl unlock and lock.
The Body Control Module is the brain of the system and when it notices the driver door is not cloed, it would ignore the locking function when ignition switch is turned on. You either need to remove the key or close the door to reactivate this function and this seems to be the symptoms as described.
Since the door ajar warning indicator is not working, it could be the BCM or ground connections that is causing the problem.