What are your high and low side pressures? Don't start replacing random parts just because people tell you to. You may just have low refrigerant which based on your symptoms may be the case. If low refrigerant, it may be enough for the compressor to turn on, but not be enough to properly cool your car. That's why it works better when moving, more airflow over the condenser means better heat exchange. So check that again, you may just have a leak.
It may be the expansion valve. If it's sticking or clogged, the refrigerant may not expand properly and this is the actual part of the cycle where it gets really cold and does the cooling.
But before doing anything, hook up a set of gauges to the high and low sides. That basically tells you everything. If the low side pressure is too low, then add more refrigerant. If the high side is too high, then your expansion valve may be clogged or stuck. If the high side isn't much higher than the low side, then your compressor is bad and not compressing.
The system is very simple. Compressor "compresses" refrigerant through condenser where heat is shed w/ ambient air. This then passes through expansion valve where it goes from high to low pressure and expands, this expansion is what causes the refrigerant to get cold. The cold refrigerant passes through evaporator which cools the actual air that comes through your vents. Then the refrigerant goes back to the compressor and gets pressurized again. So the only moving parts are the compressor and the expansion valve. Other parts can only cause leaks, they're not really going to fail.
The primary cause of bad cooling is low refrigerant. Compressors fail much less than people think but shops love changing them because it's expensive.
Been away for a while and once got back my AC died on me. I flushed the system 2 months ago and improved slightly but now the compressor isn't giving the necessary amount of pressure and only cools if driving. I was told I should change the expansion valve in the process as well.
Should I be looking at any specific brand? Saw a few on ebay.