Olds dies, then no restart. No Fault codes. Crank Sensor?

My 91 toronado will start and run fine. Then if the car is turned off, then on again, it will sometimes stall within a minute. No rough running at all, just dies. Lights and electronics on, engine not. Then you have to wait for some time before it will turn over. If you try immediately, it will just try to start but never catch. The man I got the car from said his mechanic tried everything, there were no fault codes, so replacing the crank sensor solved the fix "supposedly" a year or two ago. Would this work? Or would the faulty crank sensor not display this behavior and it is something else entirely?
Thanks.