How can I fix headlights that go out after the car is running for 10 seconds

I have a 99 Corrolla VE that has an electrical problem - the headlights, which normally are on all the time once the engine is going, turn on when I start the ignition, but go out after about 5-10 seconds. It's like a switch is being thrown. The lights don't turn on at all unless I get the engine running first - (I think they used to come on if I turned the electical stuff on with the ignition and then flipped the lights on). They always stay on about 5-10 seconds then switch off while the engine is running.

This problem was preceeded by the dash (odometer, etc) not completely lighting up. I've heard about a possible relay in the body control module (BCM), but don't know how to find the module or relay related to this problem. Any help is appreciated.