So, the car, in an electrical sense, is insane - that is what you are saying. Grab a voltmeter and measure the voltage rigth at the battery with the engine idling. Should be 13 to 14.5 volts.

Shut engine off - it should measure 12.0 to 12.8.

My bet is it won't, and that the battery is bad and perhaps the alternator is weak as well.