Many an older vehicle is subject to this same problem. Resistance increases in the headlight circuits due to corrosion and/or poor connections. The headlamp switch is usually blamed first because it has an internal circuit breaker that trips because of the resistance. Headlight switches do get old and their internal resistance can add to the issue. So sometimes a new switch can temporarily fix the problem. Most often the main issue is elsewhere in the circuit, like the headlight ground connection, and it continues to get worse to where the new headlight switch isn't a good enough bandaid.
It's not unheard of to lose the high beam switch. Since the connector was melted you would hope that was the main point of high resistance. But you really want to check the rest of the system too. Using a multimeter to measure actual resistance might show something but a better method is to look into "voltage drop" testing. These days there are cheaters with IR cameras like a Flir C3 or C2 who can just look at the electrical system with them and suss out bad connections just by the heat signature.
I seriously want one of those C3's but simply can't quite justify the price. Not yet anyway. I have been hearing about "Load Pro" leads making bad connection testing a bit easier and since they are about a tenth of the C3's price, I bought some. They should arrive next week. I'm looking forward to seeing if they are worth fooling with. The YouTube hype says they are.
Good information. But I think that a bit of clarification is in order.
First off, high in-line resistance does not contribute to the tripping of a circuit breaker. Current exceeding the capacity of the circuit breaker does that. The ultimate in high resistance (infinite resistance) would be to unplug the bulbs, right? If you do that, the breaker will obviously not trip.
Resistance that is in-line will limit current flow, but if there is is a path to ground, that would be summed with current going to the bulbs, and could contribute to a circuit breaker tripping. It should be stated that for the headlights, the circuit breaker is located in the headlight switch, and will automatically reset when it cools down. This heat up, trip, cool down, and reset scenario could be cyclic, resulting in headlight flashing.
Bad connections, corrosion or otherwise, result in resistance. At that spot there will be a voltage drop and heat will be developed. The calculation for power, in watts, is current squared times the resistance. Can the heat be enough to melt things? In a headlamp, blower motor, or starter motor circuit; yes.
If there is a path, or partial path to ground, that loss coupled with the draw from the headlights themselves could trip the breaker. There is not a lot of margin in the breaker. The most common cause of tripping breakers are bulbs that draw more current than the original design.
There was a Ford TSB on the issue in 1972 that mentioned an increase in bulb output and commensurate draw. Note the "bulb 1" document. This applies today to a bulb that consumes more power than the original.
The shop manual has the circuit breaker at 12 amps. The Cougar is listed at 18 amps - something that might be an option. But we would need to look to see if the Cougar wire gauge was increased over the Mustang - I don't want to make a recommendation that results in a fire.
If you are looking to keep things original, my recommendations are to:
1. Install headlamps that don't exceed the original bulb wattage. Shop manual lists the 6012 bulb at 40(low) - 50(high) Watts. The TSB calls out C8AB-13007-A.
2. Make sure that all connections are clean. Even if they look clean, take them apart and polish them. Look at the wiring diagram to see where all of the connections are, and clean them all.
If you want headlamps with increased wattage, install an interposing relay design, so that the headlamp switch only needs to energize the relay(s). Make sure though that the power to the bulbs is protected by means of a circuit breaker or fuse, and that the protection does not exceed the capacity of the wire. I think that I've read about a design or two being commercially available, but I have not researched any of them so can't make a recommendation.
Remember, resistance is futile!