Thanks to Bill ... which made me re-read your question. My comments (above) don't apply to your problem.
Now that I'm "focused", (and as Bill says) a poor ground (or poor wiring of the grounding of dash equipment) is a great place to start. If grounding proves not to be the problem, then looking at the "high side" (the 12VDC that feeds the gauges) is the next step.
Basically, you're looking for a change in voltage when you make a change at the headlight switch. A ground should be ground (e.g., you should not see 1/2 a volt between the grounded side of a gauge and a known-good chassis ground). A similar argument applies to the "high side" -- i.e., 12VDC measured on the high-side of a gauge (to ground) should not change as you manipulate the headlight switch. If it does, then a voltage drop is occurring ... and can be corrected when you find where the drop is occurring (usually: an under-sized wire being asked to carry too much current).
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Robi & Dale ---- Go Navy ----
2016 Winnie TOUR 42'
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or: '08 Buick Lucerne
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