1996 Lincoln Town Car brake lights not working

That last observation was very helpful. That rules out almost everything except the multi-function switch or a wiring problem associated with it. There's also an unnecessarily-complicated lighting control module under the dash, but it appears a problem with that would cause many other symptoms.

Yup. Specifically, there's a light green wire coming from the brake light switch that feeds the high-mount brake light, so we know that circuit is okay. It also feeds the signal switch, so look for 12 volts on that wire at the switch when you press the brake pedal. If you have it there, look for it coming out on the light green / orange wire and the orange / light blue wire.

It's very unlikely to find two bad contacts inside the switch, one for each side, but right before that is a single switch contact as part of the hazard lights switch. Play with and wiggle the hazard light switch to see if that gets you 12 volts on the two wires.