A slipping clutch.
your clutch is worn out.
If there were no operating clearance the clutch release bearing would constantly be in contact with the spring diaphragm causing wear on both the diaphragm and the bearing itself, also the bearing pushing on the diaphragm could cause the clutch friction plate to not be fully engaged and slip.
The clutch receives it's signal to engage from a pressure switch in the system. If there is insufficient refrigerant in the system the pressure switch will not close resulting in no power at the compressor clutch to make it engage.
You will wear your clutch prematurely. With no play, it means that there is lack of pressure on the clutch plate thus allowing it to slip; slippage causes wear.
Could be defective thermostat, bad fan clutch, insufficient fan shroud, plugged or inadequate radiator, or underperforming water pump.
I could be wrong here but are you "kangeroo jumping" as in failing to use the clutch properly when changing gears thus the car lurches, this could be a result of a worn out clutch OR gearbox.
The prefix "in" could be used to make the opposite word insufficient.
yes hydraulic could cause this also the clutch pressure plate could be collapsed
This team really pulls together in the clutch.
You could looking on individual company web pages, they often have a clearance section selling of over stock or unwanted furniture items. They often have warehouses where they only sell clearance furniture so could be a good saving for you.
If it will go into gear when the engine is off but not when the engine is running, the clutch is not disengaging. It could be the clutch, possibly warped or put in wrong, or it could be the hydraulic clutch mechanism.