1998 Saturn SL2 Stalling

First of all thank so much for your donation. When you changed your coolant temp sensor did you inspect it's connector for corrosion?If not then your going to want to do that and if it's corroded replace that connector. It sounds like you have a bad crank shaft sensor.I know it sounds weird because it only happens when cold makes it sound more like a bad coolant temp sensor but you already ruled that out unless you got a bad new sensor.I have seen cars do what yours is doing and it was a bad crank shaft sensor. Most of the time when they go bad they will cause no starts or die while driving and not start till they cool down a little. Problem with your case is it only happens for less then a min. To test them you have to perform a ohms test across the two terminals and see if there in value when your car doesn't start. You have to get to the sensor from underneath the car. So really not a option in your case. The sensors are under 20 and pretty easy to replace once you get the front of the car up and on jack stands. You also have no codes so that gives us no direction to start in when the crank sensors go bad they don't set codes anyways. So at this point you just have to try things and you pretty much tried most of the most probable things already. Would you be willing to try replacing the crank shaft sensor to see if that worked?If not then i'll try to think of something else. But I would be using a scan tool and monitoring the engine data display and recording it when cold and see if I lost the rpm signal when it died or what else was going on while it was dieing. In your case we don't have that option so that's the best advice I can think of right now is to try the crank sensor. Let me know if any new symptoms come up and what you want to do.