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OK, here's the situation. I start it up and it runs great. Drive it around, everythings fine. Next day I start it up, it won't idle. If I apply a little gas pedal it will stay running but if I let off, it dies. Swear a few times, scratch my head and walk away.
A couple days later I try it again. Start it up and it runs great. I didn't do anything to fix it, it just got better. This has happened a few times now. It doesn't run, then it runs fine.
A lot of things could cause the car to shut off and I expect you're going to get a ton of different suggestions. If you can find out for sure that the car never shuts off in this way as long as your foot is on the pedal, that helps some in diagnosis, but it may have just been a coincidence up until now.
An red-cover FSM would really help to give you ideas for diagnosing this. Go to helminc.com if you don't have one already. Also, a nice laptop scanner like Auterra would give you a live feed of all your sensors which could help isolate the problem.
Coolant Temp Sensor acts like a choke - the colder the temp, the more fuel it gets. If the signal is wrong; ie, too warm, there won't be enough fuel. Easier to diagnose with a Scanner - but that CTS temp is available through the Digital Display. The next time it won't start cold - see what it is. It should be about the same as the outside air temp. If not, you can buy a $9 replacement and without installing it, simply plug it into the harness. If it now shows the outside air temp, replace it. If it's still skewed, you'll need to troubleshoot the wiring and ECM (though if it's intermittent, I'd suspect the ECM).
ECT sensors are a simple resistor and rarely fail. It's more likely for the ECT connector to heat-warp and wiggle loose while the engine runs. Happened to me.