1992 Buick Lesabre Electrical Problem; got worse over time

Quite some time ago I began experiencing electrical oddities. The dome light never shut off so I pulled the bulbs out. The radio display blanks out and the the radio quits working when the glove box or doors are opened. When the doors are closed the radio presets are erased and the clock resets to 12:00 and the radio works again. Sometimes the headlights will not shut off. Today the car stopped running while driving down the road. The battery is fairly new and it still turns the engine over rapidly. The engine sounds like it might start but then stalls. Most of the electrical including headlights no longer works. The service engine light went on just before the car stopped running.