1993 Cadillac Deville Learching

The increase in speed is what is known as a sail on. Typically, it is a result of something being changed, or altered resulting in the idle not being learned. Had the ECM, throttle position sensor, or idle speed control motor been replaced recently? It needs the minimum air rate/tps adjustment and idle learn procedure performed to correct. If you can verify a signal on the purple wire at the starter, when it is failing, you need a starter.