Severe voltmeter needle oscillations

Here is a quick test you can do to see if the issue is the alternator.

Disconnect the or one of the field wires from the alternator.

Make sure it can not short to ground.

Drive the car ( you will be driving on battery alone so don't go too far)

If the issue no longer happens then you MIGHT have narrowed down the issue.

How long have you had the current alternator?

Would you post a photo of the voltage regulator and the back side of the alternator.