Does the cavalier have a engine shut off feature when it gets too hot

Are there any lights on the dash like ABS or SES,engine light.

An easy way to find out is to disconnect the switch. Does it shut down the motor? Typically most newer automobiles have this feature to prevent severe engine damage in the event that the oil level falls below the safe level.

is your car overheating???the fan isnt suppose to run when started(unless engine is hot), it may turn on then shut off. See if it runs when the vehicle gets hot and the A/C is on. Those engines with fans that move all the the time are belt driven. This is why they spin continually.

The starter operates great when cold. When the engine gets to its normal operrating tempreture the starter gets hot. An old starter will have an expanded armature from this causing the starter to drag or just freeze up hot . When its cool it will start again. This will eventuly get hot and the armature will go bad forever on you so get it changed soon

You may have seized the engine how ho did it get If your car is overheating and you don't shut it down Immediately or at least turn on the heater to draw the heat from the engine it will seize

This is probably a safety feature to protect the engine. I recommend having the thermostat and water pump checked

What engine?

I had this happen to me with my 91 Cavalier and it was the ignition control module. Autozone will test it for free but as you already know the problem does not occur until the engine is hot so by the time you take it in the test may not find any problems with the module although it is bad.

Ignition sensor/Crank sensor

Ive had 3 of the same car and I can tell you that it is the crank position sensor . 2.8 or 3.1 ?