Well from my experience with 70's dusters the amp meter is always live and if it goes to bad the whole electrical system can fail.
So have you checked / replaced the amp meter? Possibly explain no power unless running.
Mine was literaly cooked, burnt around the post, and it would fade in and out. I would lose all electric, motor would even die. I had to go outside the car and do the old starter relay bridge to kick the starter.
Also when you replaced the alternator did you make sure your fied wires where on the right pins. If they are wrong the alternator will over charge.
Thats just a couple things to check out.