The HVAC system doesn't use air blend doors to mix hot and cold air; it only controls the coolant flow rate through the heat exchanger. What that means is the temperature coming out of the blower will only change as fast as the water in the little radiator under the dash. If you tested the hot-cold air right after starting, you are correct, it would only blow ambient air. You really need to open the hood and look at the heater core valve located on top of the passenger side footbox. There are two long hoses and a short hose that drops into the footbox. Just before that short hose is the valve - there should be an arm with a cable on the end. The far end of the cable goes to the temperature knob on the dashboard. Rotate the knob and the cable should move the arm through at least 90 degrees of rotation.
If that works, it is possible you have air in the heater core. Go to the driver's side and lean on the large diameter radiator hose. Does it sound squishy and like water and air? Or is it semi-solid and only water moving?
If you believe the coolant is full, then find the low pressure cutoff switch. This prevents the AC compressor from running if there is no R-134A in the system. The reason it cuts off is because the R-134A carries the lubricant throughout the system and lack of R-134A = lack of lubricant. Sometimes it's called the cycling switch. Example:
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Get a paper clip and bend it into a "U" shape. Stick the two ends into the wiring plug you just pulled off the low pressure switch to bridge the connection. This will ensure the AC compressor runs when you turn on the car/AC. If it now runs, most likely the R-134A was low. If it was low enough to shut down the compressor, you can most likely add at least one can and get the compressor to stay on again.
Before you add the can, though, get a skinny screwdriver and safety glasses and a glove. Unscrew the low pressure suction port where you will attach the hose to add R-134A. Put your glove and glasses on, then poke the screwdriver at the stem in the Schrader valve. If stuff comes out, then at least you know the R-134A hasn't leaked completely out. If nothing comes out, then you have lost the entire charge and you have a big leak somewhere.
Report back.
