Instrument panel lights go out when headlights turn on

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The lights on my instrument panel (clock, odometer, speedometer, etc) work . . . unless the headlights are on. If I turn the dimmer switch all the way up until the dome lights come on, the instrument panel will light up again. For the past year or so, I had to set the dimmer switch to just less than max brightness or the panel would go dark. This new problem started last week. I checked the fuse panel, but didn't find any burned fuses.