When the engine is cold, it will stall out as soon as I take my foot off the accelerator pedal. The problems stops once the engine is warmed up.

Has the engine light come on yet?

check coolant sensor and mass air flow sensor

The choke function is not working on a cold engine.
1 clean throttle body and IAC motor air pathways.
2. Check for engine codes, you are looking for a code that relatws to the wngine temperature sensor.

the engine tickover is too slow when the engine is cold its running on cold start, ie:choke,, after it warms up the engine slows down this is what is causing it to stall in gear,, get some one to hold the brakes hard on and with the engine running in gear ajust the tick over till its reving at 750 - 850 rpm the revs will go up a bit when its out of gear,, before you pick a gear to go into push the foot brake down to stop the car lurching,,,

remove hose going to intake,clean intake plenum with a rag soaked with cleaner,do not spray into plenum because you will damage the iac motor,once you remove all the build up inside the housing it should idle-good luck

Don't let off the gas!

Check for a restricted exhaust system.Possibly catalytic converter.

I need to get a clear mental picture of what exactly is happening.

With this said, you stated that the car "runs fine" until warmed up.

Q: When you start the car when cold it idles fine?
Q: If you allow the car to idle until "fully" warmed does the car stall?
Q: As the car begins to warm up, Does the RPM fluctuate abnormally?
Q: Does the car Only stall when you press on the gas pedal or does it stall just sitting and idling in addition to pressing the gas pedal?
Q: If you can drive it while cold does the car stall during acceleration when warm or only at Stop Signs / Lights?
Q: If you wait 20min or so, can you start the car again?
Q: How many miles are on the car?

Does it have a carburetor or is it fuel injected? In 1986 it could be either way. It sounds like a carburetor with a choke problem.

many automobiles behave that way on a cold start, so you don't have to worry about it. just continue to rev the engine until it is warm!