Headlights trying to have their own dance party...

There is no headlight fuse. The headlight circuit is protected by a self-resetting thermal cutout circuit breaker in the headlight switch. It has grown soft with age and the resistance in the headlamp circuit has gone up with age, causing your problem. You could go through the headlight circuit and fix all the issues causing high resistance (new headlight switch and high/low beam kickswitch, wires, sockets, etc.) but you'll have a better result if you will install relays; see also here and here and here and here.