1997 Mercury Sable starts then dies

After car has been running for about 1/2 hour and is at normal operating temperature, the following problem occurs.
If I park and turn off motor, when I start it up again it dies. Specifically, it revs up to between 1500-2000 RPM, then steadily drops RPM until the motor dies after about 2-3 seconds.
Then, I have to wait until the engine cools off before it works properly.
Interestingly, if, during those 2 seconds when it’s running, I quickly throw it into gear and am able to start driving before it dies, it runs well. But that doesn’t always work.