Just throwing this out there ... if you can verify that the battery and connections are good, you might have a bad key sensor. It seems like your car is not recognizing that the key has been removed from the ignition. I don't know what triggers that indication (key in or out if the ignition) but to me it seems that it would be in the tumbler.

Also, maybe a reset would be good for the car. Unhook the battery and leave it for about 30 minutes and let everything reset. That might have a positive outcome.

RL