When the car is running, what does the ammeter show?
Then when the key is off but the engine is still running, what does it show?
While its running with the key off, also do some voltage checks at various locations in the feeds and ignition circuit. This, together with the ammeter
may help narrow down where the power is coming from.
I'd probably check the easy locations first.
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These with the ammeter will be clues to
a) whether the alternator is providing power or the battery
b) whether the current is getting the ignition from the ignition run (J2) or ignition start (J3) circuits.
After that I'd probably be looking at the buzzer wiring.
Also you mentioned pressing onthe ballast resistor connection produced a change.
I'd pop the connector covers off on both to look at the wire terminals.