On my 68 Barracuda with the Rally Cluster there is a separate plug-in VR for the gauges. I know this because I took it out and subbed in an Electronic one that I built from instructions in a magazine article, that has a constant 5volts output; so there is no more wandering of the needles .
The Ammeter is on a separate and isolated circuit.
To one side of the gauge, comes battery voltage.
To the other side comes alternator output.
Whichever voltage is higher, pushes the needle in that direction.
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With the engine OFF, and the Key in RUN;
obviously, the alternator is not putting out. The needle should be just a lil to the discharge side, showing that the ignition system is still on-line.
If you now have something clicking under the dash, loud enough to hear, and you have proved it is NOT the gauge-VR (which they are extremely quiet), about the only thing it can be is the signal-light flasher.
The flasher, in Neutral should not be drawing current....... unless the flasher switch wiring in the signal-switch or it's circuit, is shorted up under the steering wheel or in the column. This would put a load on the circuit and the ammeter is just doing what ammeters do, responding to it.
So then, unplug the flasher, and all your troubles should disappear. Well just until you realize ... what do I do now, lol. If that sorts it out, the next stop is to prove the Flasher is correct for your car, and if it is, then remove the steering wheel and inspect the signal switch and or it's wiring.
There is a big connector down at the bottom of the column. I have seen that connector overheated and contacts connected to things that they shouldn't be.
I have also seen the contacts in the signal switch overheated and the plastic/nylon body all melted.
If your signal switch is in fact melted, the aftermarket ones are rather hit and miss as to quality and durability. I rob other OEM columns for the factory parts I need.