It'* not on the handle, it'* on the latch. The switch on the handle is for illuminated entry. Not all vehicles are equipped with this option. However all 92-99 Bonnevilles have the dome light switch integrated in the latch instead of using the plunger type door jamb switches found in the 87-91.
The handle switch activates a 20 second illuminated entry timer within the RAC module which supplies the ground for the dome lights. If the door is opened within the 20 seconds, the domelight will continue to stay on until all doors are closed. Vehicles not equipped with the C97 option will simply illuminate the interior lights when any door is open. The wires that come from the latch are black and white. Follow them to the latch.
A broken latch switch would not cause the lights to stay on all of the time. You would have to have a short to ground on the white wire. I might suspect a headlight switch problem. Just for ***** and giggles, make sure that your dimmer switch isn't turned to the position that turns the interior lights on.