Two common problems with Dash A/C and a bit of history.
First car I had with A/C it was "Dealer installed" or after market, this type uses a thermostat to cycle the compressor on and off to control how cold it gets inside the car.. FACTORY air usually does not do this.
With Factory air the Air Conditioner compressor runs full time, if it's too cold you mix hot air in with the cold. This may be done any of a number of ways.. Air gates/doors open allowing some air through the heater core or, much more common, a water valve opens allowing hot water through the heater core, this valve may open partially, or fully depending on the type of control.
IF IT STICKS OPEN... You did a very good job of describing exactly what happens.
IN this case the air is well cooled, but it is then heated back up by the heater. (Since that is the path the air takes) Makes a great DE-humidifier though.
Page 2: NOT COOLING:
Most common cause, also the one I had to fix, listed by prior poster.. Low coolant.
Also tossed fan belt fried compressor, bad expansion valve and other system failure. Note: Tossed fan belts not common with serpentine belts and will cause other issues if they happen so you should notice swiftly.
Since I could not find my gauge... I simply added a 12oz can.. It worked.