Water coming out of the vent

You may have a dirty coil so unit shuts off on high head pressure.
Turn unit off and clean coil

If its a window unit, pull out of window get to the front fins might have to remove a few screws on the sides of front and spray out with a hose be careful not to get control box wet.spray fins and wheel inside, dump water out the back or side, put back in window it should start blowing cold air if not then its time for a new one.but sounds like its getting over heated

the thermocouple ot high heat limiter ====the thermocouple when going bad -or not properly on bracket will shut the gas valve off --same with high heat limiter ---before checking these parts feel the air coming out if feels restricted clean the vent tube under the lint trap filter can use a vacuum from front and rear exhaust tube

The window units normally have a sensor thats attached to the front of the indoor coil. This controls the thermostat. I usually pull this sensor away from the coil because it stops cooling before the room is cool. It is attached to the cold coil so it senses the coil temp, not the room temp. You can normally see it when you remove the air filter. Don't break it! It may be a wire or small copper tube

I would drain the water out of the machine via the plug on the back once the unit is drained turn on, wait a few minutes. If compressor does not come on or comes on and does not cool, you need service beyond diy.

Could be a blend door problem under the dash.

I recharge unit for your ac

does it blow blue smoke= sounds like over fueling, so you need to get the carby a service.

if not blowing smoke= *sounds like over heating. check to see if the cooling vents on the side of the engine are blocked eg wasps nests. if it has a radiator, check the water level and then look along the front and back to see if it is blocked with grass. if so blow out with hose or air blower.

You need to have an AC tech come out and do some preliminary testing on this.

Sounds like it could be a very weak run capacitor on the blower motor. Run capacitors are sized specifically to the needs of the motor. When a run cap gets weak, the motor trys to compensate for it by drawing more power. More power draw = more heat in the motor, then the motor will overheat and shut off to protect itself. When the motor cools off it will automatically restart and run until it overheats again. Keep in mind that I am a technician that normally diagnoses equipment in a hands on manner. There are many practices and concepts that transcend all air conditioning systems but it is a constantly evolving industry. I live in south eastern WA and I don't see Amana units except as window units. It is possible to have something else causing this problem (faulty thermostat, failing thermal relay(if used), problem with electronic controls(if used)) there are a lot of potential variables