I am not an expert,but i had similar problem on wifes nova. it turned out to be the dimmer switch,I would check it to make sure you are getting power in and out of it(with a test light)a dimmer switch is not that exspensive. it should be located on the steering column if you dim your lights from the turn signal switch. Hope this will help you out.
Turning on the headlights with the engine turned of.
Headlights run off the battery not the ignition...
Because Mitsubishi Eclipse 1996 doesn't have automatic headlights.
Turn the headlights on the whole way, then push the headlight knob back in, leaving the parking lights still lit. This will hold the headlights up, without having them turned on.
Headlights might stay on in a Honda Odyssey after the engine is turned off because of a courtesy timer. The headlights will stay on for a few seconds after exiting the vehicle so the driver can go into a building at night safely. If the headlights are staying on all the time, check the timer for this feature.
It is not supposed to be on when engine is running unless there is a problem.
have you tried turning the ouside dial on headlight switch that dims the gages bright or dull had a similar problem with 98 corvette no dash lights or hvac or running lights turned out to be multi function switch (headlights turnsignals)
I am having the same problem with a 1994 model. The high beam indicator is also on. I believe it is related to the daytime running lights because when I turn on the headlights the lights get brighter. As I understand it daytime running lights are the highbeams at about 50% power. I need to find where that relay is and test/replace it. wish me luck...
check the battery and connections/cables
Its not your engine. Its the fan running to cool the engine down. They all do it and nothing is wrong with your car.
use your parking brake this answer is in the owners manual
sounds like a bad light switch.