If I am understanding everything to this point, the car starts and runs pretty much as you think it should. It idles fine. When you step on the gas the motor speeds up, when you release the motor comes back to idle.
Assuming for the moment, if you turn on the head lights and the a/c if equiped nothing out of the ordinary happens?
It still continues to run correctly.
The problem only occurs once the transmission is placed in either reverse or drive.
Again, assuming it is automatic which it appears to be from your description.
And actually, once in gear and moving, the car actually runs o.k., it is only when you come to a stop, try to come to a stop and you are almost stopped or you are stopped already and try to put the car in gear. It makes no difference whether it is forward or reverse, one the car is almost stopped or stopped you have to play with the gas pedal and the brake pedal to keep the engine running.
Does this pretty much sound like what is happening, if not, where am I wrong?
You have had the computor scanned for codes and there are non being displayed. So as far as you can tell, there is basically nothing wrong with the car other than when you try to come to a stop or the car is stopped and you place it in gear, the engine comes to a halt and stops running?

 

Yes this describes the problem. It doesn't do this when the car is cold, it only starts doing this after it is warmed up. But you got the problem down exactly right.

 

When the engine is cold it requires more fuel than normal to keep it running. A fuel injection computor controled engine does this by adding additional fuel through seperate curcuits untill such time the engine warms up.
Then those things shut down and are not used again.
The symptom as described could be any number of items, some simple, some complex.
Complex ones have a tendancy to leave a Code trail that can be monitored so simple sounds better.
Vacuum leak, a bad rubber hose, cracked intake manifold, some thing of that nature.
Plugged fuel delivery, fuel is delivered through the injectors, there are one for each cylinder. They probably are working, pressue should be o.k., would be a good idea to measure it, the port to measure it is on the fuel rail.
A dirty Throttle body full of carbon and other yucky stuff could plug up the air delivery under warm conditions.
These are things that should be some what obvious to those who have done some mechanical work at all, but are easy to over look.
The only other thing that comes to mind to me and it is something I have run accross, and that is in the transmission. There is an item called a TorqueConverter, newer types are referred to as a
lock up torque converter,
It has a clutch in it that cuts down on slippage when driving. The torque converter is supposed to slip that is the way it works. I lets you change gears when stopped. The clutch locks it up at road speeds to increase fuel mileage and decrease heat being genrated.
If that clutch does not release correctly than it will drag the engine to a halt when the car comes to a stop.
This is not a usual thing to happen only when the car is hot and not cold.
It is something that when it is a problem it is all the time
or
I would suspect that it would happen more often when the transmission is cold and valves are sticky
but truth is stranger than fiction
and some cars were prone to that at various times in thier lives.
I have not heard of Saturns being prone to it,
but
you might have the one.

 

awhile back my 96 saturn sL1 was starting to drop on RPM"s too
but it never stalled however being a 96 with over 260,000km"s i suspected a dirty carboned up throttle body and to my surprise
after checking it yeah..! it was carboned up black! in there went down
to my local parts store and bought a can of throttle body cleaner
and used an old toothbrush and cleaned it right there on the engine
and that"s what it turned out to be i would try that 1st and see how
that turns out for you??