On my 1969 Mark III the headlights will flash wildly and there is a weird buzz that occurs when the headlight dimmer trys to change either high to low or low to high. The headlights come on fine when first turned on but drive 3-4 minutes and the flashing begins. I no longer drive it after sundown.
I assume that you have the Automatic Headlamp Dimmer option. If so it sounds like the Photo Amplifier unit is faulty. It is located outside on the cowl near the windshield. I would try unplugging this unit and then operate the headlamps to see if they then operate normally. If so you may need another unit as these are non serviceable. There is also a Power Relay unit in the circuit but your description sounds like a Photo Amplifier issue as discussed above. Have you tried operating the headlamps with the beams locked on low position. This as you may know is operated by the foot dimmer switch that is located at the left floor area near the left air vent outlet and is visible in the carpet. Pressing this switch with your foot will switch the beams from Hi to Low and you should hear no buzzing or experience any automatic Hi /Lo beam switching when in the Lo beam position. The Autodim feature only operates in the Hi beam position. You also don’t say if there is a reaction when you adjust the sensor dial behind the headlamp knob as adjusting this this may help or change the situation. At Lincoln Land we would however try another relay and if that fails to correct the erratic operation we would then try the Photo Amplifier. If you have more information after trying the adjustment or would need any parts or further advice please contact us at any time. The procedure for troubleshooting the Automatic Headlamp dimming is provided in the Shop Manual. This manual would be a benefit to you.