the voltage should be the same on each pointset for all intents and purposes.
One of the coils is bypassing the ballast resistor, mismatched coils perhaps, bad bullet connector on black/yellow or black/white, bad wire.
Cold, the points should get around 12V. As the ballast warms up the voltage will drop off relatively evenly to each pointset. The ballast warms as soon as the power is switched on and the points close. I think.
By "sat awhile" I take it you mean the ballast is cold to the touch, so if when cold you only have 6V to the points I would measure the coil primary resistance, s/b 1.7-1.9 ohms or thereabouts for the stock 6V Lucas coil. Check the coil end for the Lucas number for identification if in doubt. If that's all good then look at the wiring for damage and connections for proper routing.
Was running correctly before what?
It should still run correctly as is. One of the coils will get overly hot, the points may pit more quickly, you will have spark unless your problem is a big resistance in the circuit.
I bet if you check the voltage when hot the 6V you see initially cold will go to almost zero and then no sparky. Check for good connections.
I need some sleep.
All the best.