The fan discharges into the evaporator coil. If air is flowing out of the vents, it's flowing through the evaporator.
If the blend door is stuck on full heat, then it is possible the cooled air is being re-heated. One way to test this is to disconnect the heater hoses and see if it cools down. Another way is to let the engine cool completely, start it and run max AC. If it starts blowing cold air and warms with the engine warming, then it could be the blend door.
Look down at the A/C clutch on the front of the compressor. Is it spinning? It should be. It should not cycle more than ten times per minute.
Feel the orifice tube, it should be icy cold. It's the skinnier of the two lines that go through the firewall, the one that does not go through the dryer.
You might have a plugged orifice tube, a bad pressure switch, low coolant, or worn out compressor (will not make enough pressure).
At the end of the day, you need to get a manifold gauge and look at the high and low side pressures.
Good luck.