Trying to set up a pump without a pressure gauge is like a blind man trying to drive, or like trying to find a problem in an electrical system without a meter.
How big is your tank and how many GPM is your pump? What is the time to fill the tank (start to stop) with no usage?
The short cycling suggests that you have a problem with the air tank or the pressure switch settings. Symptoms are as if the zone is connected somewhere different or that there is a check valve somewhere. The pump should have reasonable cycles with NO zones on, and adding a zone should not "short cycle" it.
Procedures to use:
1. Buy and install a gauge near the tank and pressure switch (They are close together, I hope.)
2. Verify that the pressure gauge and the tire gauge that you are using to check air in the tank are reading the same, or nearly so. Do that by measuring air pressure when you have a gauge reading on the water gauge with some water in the tank. They should be the same.
3. Verify that your pressure switch line is not clogged with rust or dirt.
4. Verify that your pressure switch is not connected to a small tee that carries water to the last sprinkler zone. The pressure switch and gauge should be measuring tank pressure; not pressure in a high velocity line.
Fix the cycling problem before you ruin your pump and another pressure switch.