I went to the junkyard and snagged a switch from a '96 Bonneville the other day and installed it this afternoon. Everything went smooth and I have dash lights once again! As predicted, several contacts on the old switch looked burnt and the connector on the wiring harness was melted.
Interestingly, the dash lights seem brighter now than they did with the old switch. I had always thought they were a bit too dim, but put it down to getting used to the red dash lights. The Twilight Sentinel still doesn't work, but I guess I can live with that.