Shaking while braking

I DON'T KNOW IF THIS WILL HELP BUT NARROW IT DOWN,WHEN BRAKING WHAT SHAKES,STEERING WHEEL SHAKES BACK AND FORTH IT WOULD BE A FRONT BRAKE ISSUE,MAKE SURE CALIPER HARDWARE IS OK,AND CALIPER SLIDE PINS ARE FREED UP,IF THE WHOLE CAR SHAKES IT'S THE REAR BRAKES,PINS SLIDER AND HARDWARE ON REAR ALSO,HOPE THIS HELPS

ABS brakes do pulse. If that is not what you are detecting you can get the error codes. Many Auto parts stores will have loaner meters for free. Cadillacs have lots of sensors.
Anti-lock brakes have sensors to control braking which could be the problem.

If this car has a StabiliTrak chassis then it will have a continuously variable road-sensing suspension with a fast-acting variable dampers, integrated rotary sensors to measure body and wheel positional changes due to varying road conditions.

This car does have recalls you can check.

APR 17, 2000 | Recall ID# 69145

SERVICE BRAKES, HYDRAULIC:FOUNDATION COMPONENTS VEHICLE DESCRIPTION: PASSENGER VEHICLES. SOME OF THESE VEHICLES HAVE INTERNAL FLUID LEAKS IN THE BRAKE HYDRAULIC CONTROL UNIT

Have you tried getting the rotors cut. they will warp over time.

Last resaults try changing all the break fluid get it done right with the BG44 PROFESIONALS Problem: Brake fluid becomes oxidized and contaminated over time and use. Heat and moisture cause corrosion on parts and brake fluid failure. This causes a spongy brake pedal and shows up in discolored brake fluid. Solution: The entire brake system is inspected for leaks, master cylinder corrosion, worn pneumatic parts, harmful varnish build-up, broken or rusted bleeder valves, worn rotors and drums and air in the brake lines. All old brake fluid is replaced with BG Heavy Duty DOT 3 or DOT 4 Brake Fluid. The result will be a longer lasting, more efficient and safer braking system. http://www.bgfindashop.com/bgservices/brake.htm