89 Chevrolet Cavalier 5 to 10 miles cut off but start back up pull in gear cut off again start thier about 5 mins start back up run fine to the next stop and cut off again

SOURCE: hesitation

I had the same symptoms. First off, how long since you have replaced the spark plugs and wires? I know they are supposed to be 100,000 mile life but not in my case. Here is what I did to correct my issue (note: it did not do it all the time, only occasionally and seemed worse when it is cool in the mornings)
1. Replaced Spark plugs, the issue got better but did not go away.
2. Took throttle body off of truck and cleaned all the carbon from around the buttefly with toothbrush and carb cleaner. Took idle control module out as well and cleaned carbon as well, issue got even better but did not go away.
3. Replaced GM spark plug wires, only 55,000 miles on them, one at a time (just to determine which one was causing the issue if it was a plug wire). Cylinder number 6 wire was bad. So, replaced plug wires and wa-lah, problem solved. Not saying that is your issue, but it corrected mine.
4. While I was at it, I replaced O2 sensors b/c I did have one that was lazy according to the computer.
Truck now has 231,000 + miles and runs like a champ.

SOURCE: Truck loses power sometimes.

A dirty or defective mass airflow sensor will cause hesitation issues also clogged cat what code did ce light put out?

SOURCE: 95 lumina dies and then starts back up and runs fine.

Electrical problem can be the most frustrating and difficult to diagnois especially when you are not phsyically at the vehicle. I will attempt to point you in the direction you should be looking. Right off the bat it sounds as if you are running low on power when the demand is high. First thing to suspect is the battery. For some reason it is low on electrical power.
Next thing is to figure out why. The first thing I would check are the cable ends. Are they corroded? It may not look as if the are, but they still could be. Clean them thouroghly.At times, when they are corroded that may be enough to stop the flow of current from the alternator to keep the battery charged up. If they look good, you may want to put your battery on a charger. Pay attention to the meter and see how much the battery is down. If it taking a lot of charge it could be it needs replacing, but charge it up and see how it works. Remember when charging a battery, first remove the caps and see if the cells are low on water. If they are your battery is probably the culprit. When a battery starts using water it's usually because some of the plates have over heated and warped, which causes them to touch each other and short out. BTW, when charging a battery make sure you remove the caps during charging. So leave them off after you check the water, AND if you need water use only distilled water. You can buy it just about anywhere, grocery store, Walmart, KMart, etc.
Check your altanator belt and make sure it is tight and not streached out. That could also be why the battery is not keeping up with the demand.
If that dosen't cure it, next take your battery in and have it tested and see if it is up to snuff. I hope this helps you and please let me know through your rating of this advice.

SOURCE: Cavalier A/C isn't working.,

If you see wet spots on the compressor it would be an indication that the compressor is bad, however buy a can or refrigerant with leak dye in it and see if you can see where the red dye is comming out and that will be your answer. Rather than guessing which one it is check it that way so you will know what it is then I can tell you how much to fix it.

SOURCE: 89 z24 stalls instantly when shifting out of park

Your lockup torque converter is bad I believe. When you shift from Park into any gear it will die.