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.

Thank you so much for posting the above situation and the subsequent response!!
I have a 1998 Saturn (82,000 km) which displayed symptoms similar to the above; it would start easily in the morning and at the end of the workday, it would run well for about 5 minutes then warning lights wOuld flash briefly and sometimes it would buck and it would turn itself off -sometimes while driving (and I would coast to a stop) and sometimes at a stoplight. After this type of episode, it was difficult to restart.
I took it to the mechanics twice and their computor indicated nothing wrong. They drove it four times and the car behaved well for them.
For me the car continued to stall out. I googled and found the above situation and recommendation.
I called the mechanic again - told them the car continued to stall for me and it was getting worse. I requested they replace the crank shaft sensor; They stated they would consider it and to bring it in again......
My car finally displayed its symptoms in their Shoppe and the mechanic was able to capture the situation on their computor.
They replaced the Crank Sensor and the Idle Air Control Motor.
Since then, the car has been behaving beautifully!!! (and I am pleased I do not have to think about getting a new car yet)
HAPPY NEW YEAR!!!