The refrigerant is not supposed to be made "completely full" rather a charge is measured by weight. Failing that, the other accepted method is to charge, and observe the behavior of the low side and high side of the refrigeration system. That requires two gauges to do properly.
The system relies on the fact that condensed, liquid refrigerant needs to "boil" in the evaporator in the car, to take heat from the air. If there is too much coolant in the system, it cannot "boil", and it cannot condense to a liquid.
I am not confident with the OP observations. Normally a system "completely full" would stall the compressor or the compressor would not start.
So here are some questions:
- When running the AC is the compressor turning (and do you hear the clutch engaging so that the belt pulley turns the shaft of the compressor)?
- How long does the compressor run? Continuously, or for short times like 5 seconds or so?
- What is the ambient temperature, and what is the temperature of the air coming out of the vent in the vehicle?
- Is the engine cooling fan running when the AC is turned on for the car, and does it stay running when the compressor turns on and off?
Unfortunately, a complete answer is not possible given your information, and it is necessary to get some questions answered first. Alternatively, a complete answer would be a book (grin).