Electrical Gremlins

[Frankenstang65]

So here's the deal.... whenever my car (65 coupe~351W) is running all of my lighting accesories seem pulsate, getting bright and then dim. The battery is new and the alternator looks to be in good shape. I'm thinking that it may be the constant voltage regulator but am not really sure. Also... my instrument panel lights do not turn on when I turn on the headlights/parking lights. All the wires are connected and the bulbs are new. Is there something wrong with the switch that only affects the cluster lights? Could it be a fuse? I am almost finished with a full body and interior restoration on the car and would like to take care of these little annoyances before I get her all back together.

Any advice would be very much appreciated.... Thanks