Posted this on another post with same problem. Hope it helps.
Don,t know if this helps but I have a 94 cavalier in shop right now to have fixed and one of the problems is the same as yours. My mechanic says it may be a bad ignition switch or on my 94 it has an ajustment under the dash on the steering colomn that sometimes gets out of ajustment and has to be reset. he said he noticed that when he turned it over and did not let the ignition switch go all the way back it tryed to stay running. I don,t know if your car is set up the same way but maybe someone else knows if it is. Hope this helps.
fiserman