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97 98 99 Bonneville headlight switch
I finally decided to read the owner's manual (imagine that) to see what it had to say about the headlights. It was only after I did read that information that I had an idea about how the headlights were supposed to work.
All of these cars have the Daytime Running Light (DRL) feature, whereby the headlights come on about half bright on low beams as soon as you put the car in gear. When the light outside gets below a certain level, a sensor on the instrument panel turns the headlights on full bright low beam. If you are in a full-light situation where the sensor has the headlights in the DRL mode, you may turn them on full power low beam using the plunger in the switch. If you are parked and the headlights are off, you may use the plunger to turn the lights on and off; otherwise the plunger isn't used much.
I never use the Twilight Sentinel feature (switch under plunger knob that if turned on, will leave your lights on for a certain period of time after you shut the car off to give you time to get into the house using that light).
So, my headlights weren't doing what they were supposed to. It occurred to me to do something simple. Clean the dust out of the little dark blue eye on the light sensor on the left upper side of dash. My wife uses this car and it's always clean inside, but nobody has ever thought to dust the sensor. I just ran a Q-tip around in the little eye a few swipes. The next day, the headlights were working just like the owner's manual said they should. No lights on with car running and in park; come on half strength (DRL) when put in gear; and come on full strength in shade or darkness. Instrument lights come on bright with DRL on; dim when headlights are full on.
Somewhere along in there, I went to the junk yard and taught myself how to change a headlight switch on a 99 Bonneville SLE which is just like my 97 SSEi with respect to the lights. This was a "just in case" so I have a spare switch now if I ever need it (but probably won't).
In case you ever need to know, the headlight switch comes out straight toward you in the driver's seat; it's impossible to do it from under the dash like an old Ford. You have to remove the dash knob (little ring under the plastic that pushes in to release), pull the Twilight Sentinel switch straight off, take off the plastic trim above the instrument plexiglas (three screws), unplug the cigar lighter, and three Torx screws hold the h/l switch in. It's a very tight fit getting it out, but once you do that, carefully unplug the wiring pigtail and Bob's your uncle.