Engine stays running without key in ignition (and more!)

Went back underneath, and noticed the ammeter wiring was wedged between the light switch ground post and one of the ammeter posts. I relocated the ammeter wiring away from the switch ground, and now the dash lights work properly again.

Someone tell me if this makes sense: when the positive battery cable melted and grounded against the header, it shorted and caused the ammeter to spike
Yes.
Starter cable grounding to header caused the alternator to supply as much power as it could. The ammeter maxes out at 40 amps. Orange arrows show the path of current flow.
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the ammeter wires got hot and melted against the light switch ground,
Reasonable theory.
It's likely that wiring was eventually going to be a doomed situation regardless. here's why I suspect that. As far as I know, the factory light switches did not have a ground tab. Only thing that needs a ground in the headlight switch is the dome light control. Usually grounding for that was just the attachment to the dash. I think the headlight switch was replaced and at that time the wires got pinched. The factory routing in general was pretty good in providing support for the electrical cables. Mechanics (like us) often find wire routing in the way and/or a pain to undo and put back so we don't bother. Works when you only keep the car for 5 more years. Not so with 15 or 25 more years!


shorted the switch, which caused the switch wires to melt together and kept the dash lights on whenever connected to power.
Something like that. As far as you can tell the switch is fine. So seems more like the insualtion between the instrument light wire and one of the power supply wires got chaffed/cut/pinched and melted together.

Neither of the wires circled appear to be the power feeds to the light switch.
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If either wire connecting to the ammeter grounded, the fusible link would have melted.

The feed to the light switch B1 terminal from the main splice probably is black with a white tracer.
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There is also a second feed to the light switch coming from the fusebox hot side connecting at the B2 terminal.
Colors and routing here are for '67 but concept is the same
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B1 powers only the headlights. Internally there is a 15 amp circuit breaker.
B2 powers the parking lights, and alos goes to the rheostat for the instrument lights.
Wire for the Instrument lights then goes back to a 2 or 3 amp fuse then to the instrument lamps.
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