Hi Rob,
Everything that has been said by the others is correct.
Your old switch melted because of a short circuit.
From your pictures I noticed that you still have your center console installed. You must be a contortionist !
The center console is easy to remove and will give you better access to the switch and wires in the dash.
The short could also have been just the switch going bad after
40 + years. You said you don't know much about electricity so
here is some trouble shooting for beginners. Turn off the big disconnect rotary switch next to your battery in the trunk ( boot ) before you start doing anything.
1.. Order a new light switch.
2.. Remove your center console so you can get to the wires
easier.
3.. Make certain that the metal part of the wires are still firmly attached to the connectors that have the melted plastic insulation. If any have detached from the connectors, you will have to obtain some new slide on connectors from a hardware or home improvement supply store and install new connectors on to the melted ends of the wires ( after trimming off all of the damaged wire ).
4.. If the metal part of the wire is still firmly attached to the slide on connectors but the insolation has melted away exposing the metal part of the wire, you will just need to insulate the wires were the insulation is missing. This can be done by cutting off the damaged portion of insulation ( be careful not to cut the metal strands of the wire ), and then use either heat shrink tubing or rubberized cloth type of friction tape. I would buy some pieces of heat shrink tubing big enough to slip over the ends of the slide on connectors and some of the tape. Don't use the plastic electrical tape that is commonly sold now. It will dry out and drop off after a few months.
5.. Now that you have your connectors fixed and insulated again, hook up the repaired wires to your new switch. Let the switch dangle loose under your dash and heater. Just make certain that none of the wires will touch any metal and short.
6.. Turn on the big switch in the trunk.
7.. Turn on your light switch one click to the parking light position. Quickly look to see if all four of your parking lights are on as well as the license plate light. If not, turn off the light switch and rotary switch. You will have to find the short in your RED wire circuit that feeds your four parking lights and license plate light. If just one or two parking lights are off but one or more is on, turn off your light switch and then you will have to trace the red wire to find the place it is touching the metal of the car ( ground ).
If just the license plate lights don't come on, look at the wire were it comes out of the hole in the trunk. That is a common place for the wire to short through the insulation and cause a problem like yours.
8.. Once you verify that all four parking lights are working, turn your panel switch on and then turn on the light switch again just one click. Check to see that all of your dash lights come on. If not, look for the problem with the red/white and black wires that connect to all of your gauges.
9.. Once you verify that all of your parking lights are working as well as your license plate light and your dash lights, leave them on and touch the sides and connectors on the light switch. Is anything HOT to the touch ? If so, you still have a problem somewhere in the red light wiring so you should seek another person that is knowledgeable with electricity or Healeys or both to help you find the problem.
10.. If you reach this point and everything appears normal and the switch didn't get hot, turn the switch all the way on to the headlight position. Quickly see if both headlights come on. If they both come on, touch the switch again and see if it is getting hot to the touch. If yes, turn off the lights and seek help finding the problem in your headlight circuit. If the headlights don't come on at all feel the switch and see if it is rapidly getting warm, if so, turn off the lights.
11.. If the switch is NOT hot to the touch but the headlights are not on when the switch is turned on, push the dimmer switch once. Did the headlights come on or does the light switch start getting hot to the touch ? If yes, the dimmer switch could also be bad as well as the wiring to the headlights.
12.. If the headlights work in both high and low beam mode. Let all the lights stay on for 5 minutes or so and then feel the switch again. Is it HOT now ? If not, then perhaps the only thing that happened was the original switch went bad and shorted after 40 + years of service.
..... You should get someone to help you that knows something about electricity if you get stuck trying to find out why one or more of your lights is not lighting up or if your switch gets real hot, real fast when you turn on the light switch.
.... If the light switch gets hot to the touch quickly after you turn on the light switch, then turn everything off and get some help.
:crazy: