I didn't have the exact same problem, but last year I had my dash lights go dark while driving at night. The cause was the infamous dimmer switch.
Thanks for the data point.
I double checked the dimmer switch, and it works fine with running lights only. But with headlights, they go out completely.
I've been pouring over the wiring diagram. Right now, I suspect it's the either internal to the headlight switch, or the connector on the headlight switch.
But I've had enough for the night.