Sounds like a fault with the Antilock Braking System. The vibration in the pedal is the ABS working. With your foot on the brake pedal, it releases and re-applies the brakes to stop a skid from the wheels locking up. You need professional service for brakes.
Seems to be a known issue on all Cadillac and requires the ABS module to be replaced.
repairpal.com > Problems > Cadillac > DeVille
Cadillac DeVille ABS Light Due to ABS Control Module Failure. This is a verified problem based on 126 reports. Learn more about this problem, when it occurs ...
repairpal.com > Problems > Cadillac > Seville STS
Traction control and ABS lights stay on dash. This is due to ecbtm Module gone bad. replace Module and this should fix the problem. You can go by the VIN and ...
repairpal.com > Problems > Cadillac > DeVille DTS
ABS Light Due to Control Module Failure on Cadillac DeVille DTS. The anti-lock ... ABS and Traction control light stays on. ... abs light to abs control stay on.
repairpal.com > Problems > Cadillac > Seville SLS
ABS Light Due to Control Module Failure on Cadillac Seville SLS. The anti-lock brake ... SLS135,000Jumsee. Abs and traction lights stay on when started.