Hello. W/D here. There are a few things that can cause your issue, and we'll go through them one at a time, hitting the most common first. Make sure that your pressure gauge is good.
First, let the pump kick on and go to its maximum (kick-out) pressure. Close the discharge valve, between the tank and the house. Watch the pressure gauge at the tank/pump. It should remain stable. If the pressure falls over a period of 10-15 minutes, look for any obvious leaks and repair them. Retest. If the pressure falls with no signs of leakage, the problem is within the well itself, usually the foot valve leathers or check valve. The easiest fix for that is to remove the pump, securing the well pipe, then removing the internal well pipe, and changing out the bad components at the end of the inner well pipe. My best advice is to change out everything as the components seem to be on about the schedule of deterioration, and it's easier to do it well once than to have to pull it again later for another component going out. The defective parts can include the foot valve, deteriorated jet valve (common), and deteriorated leather/neoprene seals (most common). Changing all of the components at once isn't expensive, and a local well installer can check the assembly for you and replace the defective parts in their shop. My local installer takes the assembly apart, inspects and rebuilds it, charging me only for the parts. Don't forget to get gaskets for re-installing the pump. I keep spares of the leathers and pump gaskets in my home shop, as wells and pumps always seem to go out on Sundays.
Provided that the first pressure testing went well, open the valve to the house and watch the pressure gauge. If it falls, you may have a leak on the house side of the valve. Check first for leaking faucets, toilets, and items that may require water (icemaker), etc. Isolate these and retest, if necessary.
Go back to the tank. Check for the location of a schrader (bicycle tire) valve on the top of the tank. If you have one, then you have a tank with a bladder in it, providing the necessary air cushion to move water when the pump is not running. The directions for checking the pressure should be located near the valve stem, but, if not, I'd set the bladder pressure at 28-29 psig, with the pump turned off, and the tank depressured. You'll need an air compressor or bicycle pump and a tire pressure gauge to set the pressure. With the tank depressured while making the bladder pressure adjustments, there will be an open valve on the tank (the one that you opened to depressure the tank). If you hear air coming out the valve as you are adding air to the bladder, you may have a defective bladder. When you air it up, check the pressure, wait a few minutes, and check it again. Loss of pressure indicates a defective bladder, and the tank-bladder will have to be replaced as a unit as they are sealed.
IF there is no schrader valve on the tank, there is another type if system, less common now, for maintaining an air cushion in he tank. It is an air injection system, where a tiny amount of air is injected each time the pump cycles, maintaining the air cushion in a manner similar to the bladder system. Sometimes the injection system can get out of balance, and too much air can get into the tank, causing a condition known as "air logged". Usually, the water delivered to the house will make a spitting noise at the faucet giving you water and air at the same time. The easiest way to identify this type of system (air injection) is to look for a component on the side of the tank (sometimes, if it is plastic, you can see a ball in it), and tubing will go from the pump to the injector. The quick fix for this type of system is to vent off some of the air from the top of the tank. If you can see a "sweat line" on the tank, indicating the water level, you can "guesstimate" the level in the tank. The air cushion should be somewhere in the 1/4 to 1/3 of the way down from the top of the tank. Bleeding air off of the top will decrease the air volume in the tank.
If everything checks out well, you'll need to check the pressure switch. Turn the power off to the pump motor (usually tripping the breaker at the box, or tripping/pulling a "service disconnect"). Once completely isolated from any power source, Close the valve to the house and depressure the tank again. Remove the cover of the pressure switch (it is located right at where the power comes from the breaker to the motor, and passes through the switch first). The switch proper usually looks like a little box, with the power line coming to one side, and a small water tubing line connecting to the bottom of the assembly. There will be a nut on the top of the switch. Remove the cover, and inspect the contacts of the switch without actually touching them. If the contacts appear to be burned, or any of the wiring appears to have been burned or show signs of having been overheated, the switch must be replaced. These are inexpensive and most larger hardware stores will have them, or you can get them from a well service company. Make certain that the power is dead off before disconnecting the power, and make a sketch of the wiring to the switch terminals before disconnecting the wires. Make sure that the water pressure to the switch is depressured, or you'll get wet.
If you have an air volume switch as we discussed earlier here, and not a captive air type system with the bladder in the tank, you should consider going to the bladder type in the future. The bladder type is more reliable, and takes a lot of the troubleshooting out of the equation, making the system easier to maintain.
Anyway, these are my thoughts in the issue. Hopefully, they'll get you sorted out!
Best regards, --W/D--
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Sounds like you have a waterlogged tank. If it is a tank with a bladder in it you might have to replace it. Does it have a stem on top to put air in? if so it is a bladder tank. One thing to try: turn off the circuit breaker, open the hose bib close to the tank and let all the water drain out, when empty check the air in the bladder with a tire gauge should read at leaset 20 psi. Add air if needed, close hose bib and turn circuit breaker back on, if tank is leaking this should help temporaly, if it starts doing the same thing in a short time you will need to replace the tank.