Why won't the brake lights cut off?

There has to be a switch up there somewhere that is out of adjustment. The spring could be weak too, but if the brakes aren't staying applied while you're driving, the pedal has come back far enough.

Look for a gold threaded shaft. There's either a nut that's loose or two nuts to set the position, then tighten 'em. Would it help if I posted a photo?

There's two versions depending on with or without cruise control. The other one has a silver metal body.

First you have to see if it's loose. If the nut has loosened, the switch will push back instead of the button getting pressed in by the pedal arm.

If the switch is tight, it may need to be adjusted but you must understand that over-adjusting it can hold the brake pedal down a little. That will make the brake lights turn off but it can keep the brakes applied a little. You'll notice that when driving for a while when the car seems to be moving hard.

My money is on the nut being loose because those switches don't magically get out of adjustment by themselves.

On the brake pedal where the switch and pedal brakcet meets, ensure there is no hole on the bracket, meaning a stopper is missing and this would result in the brake switch being turned on at all times.

What was the fix for the brake light switch?

I'm not familiar with the specific circuit for the dash lights, but if only some of them are out, 99 percent chance those bulbs are burned out. If all of them are not working, many cars by 1996 had stopped using the simple, reliable dash light rheostat and began using "pulse-width modulation" dimming which means a relatively complicated electronic circuit to do a formerly simple job. Often those dimmer controls are mounted separately in the dash rather than being built as part of the head light switch. You can likely find one in a salvage yard to try rather than buying a new one, then finding out that isn't the problem.

Ensure the brightness control switch is not turned to minimum. Try twsiting on it to see if the light comes back on.

If the problem was not there prior to working on the stoplight switch, ensure the brightness control switch wire connector is not dislodged.

It should be somewhere around the meter cluster on dash.

Behind the switch.

The switch is a rheostat. Disconnect the connector and use a 5A fused jumper across terminal 2 and 4 (Black/Yellow and Green/White)and turn lights on. If dash lights comes on, the switch is bad.