Yes, the larger pipe has the low side port. If it isn't getting cold, that is telling me that either the refrigerant isn't flowing or flowing too much. Is the small pipe hot? Either way, it sounds like your problem is not the temperature door.
With the AC on and engine running, the hub of the AC pulley is turning with the pulley, correct? If so, you will need a set of pressure gauges to see what is going on. General rule of thumb I have seen with most ac systems, with ambiant temperature about 80-90°f, the low side should be about 30-40 psi and high side about 200-250 psi.
If both are low, then the charge is usually low. If both are high, then the system is usually over charged. If the low side is low, high side normal or high, usually a restriction on the system. Low side pressure high with high side low or normal usually indicates an expansion valve stuck open.
If both the high and low are close to the same, usually indicates a failed compressor. Fluttering needles on the gauges, usually happens on the high side, indicates failing valves in the compressor.
If the hub of the compressor pulley is not turning or starts and stops frequently, you have either a pressure switch or electrical problem.
All this assumes that your AC has a proper charge +/- 1/2 pound of 134a refrigerant.