I have been thinking about this for a while and it had me stumped for a bit.

normally power is only applied to that circuit from the key in start.

however, once in run, it could back feed from the ballast back to the ignition.
now normally this isn't an issue because it could not complete the circuit since the ignition is in run, and even if it were not, there should be no path to ground since it is the hot side of the circuit.

But if one of your connectors is not right and has a partial ground going, enough to heat it up and make it act like a bimetalic, that might explain your symptoms.

While it might not be the switch itself, how does the connector look at the ignition? or at the column? how much of that is new?

If I am right, you don't even have to run the car, just set it in run for however long it would normally take to cause this to happen.
It is also possible to test in run and not bother with start, as you will see power up till the break from the back feed.

again just a guess, but the only one I can think of. one of the wires might also have an insulation problem, something with a wear in it so only 1 strand is touching the coulnm not enough to blow a fuse, or melt the line, (not even sure if it is fused).
since it only gets 12 volts on start and the backfeed would be the 5 or 8 volts to the coil.