Sounds like to much voltage drop due to the draw the f the headlights. I'd first check and clean connections and then evaluate if the cigar lighter is connected and grounded at ideal locations to minimize effects of voltage drop from other loads.

•1951 GMC 250 in the Project Journals
•1948 Chevrolet 6400 - Detroit Diesel 4-53T - Roadranger 10 speed overdrive - 4 wheel disc brakes
•1952 Chevrolet 3800 pickup
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"First, get a clear notion of what you desire to accomplish, and then in all probability you will succeed in doing it..." -Henry Maudslay-