Yes, you've got an electrical problem of some sort. My Cobalt does something
similar in damp weather if its been sitting unused for a day or two. I "fix" mine by
switching the lights to high beam for a second or two. My guess is that a molecule
or three of moisture is finding its way into an electrical connection somewhere and
interfering with the flow of electricity. The extra power demand of the headlights
probably creates enough heat in that connection to make the moisture evaporate.
Just a theory, probably wrong, but its all I can think of. They're old cars and not
worth spending hours or hundreds on to chase something that's doing no harm
and is easily corrected by applying the brakes. But if it bugs you and you've got
nothing better to do, then yes, go through the electrical system. Clean and tighten
all battery connections, grounds, and what ever electrical connections i the wiring
harness you can find. Hopefully that will make it go away.