The control oil pressure is generaly used for operation the drive clutches and pressure lubrication of bearings etc. Now I have not worked on the one you have, so don't know what the air would be used for. Maybe a holding brake when the transmission is in neutral? You need to get a manual for that machine. But in my experience you have an unusual problem.
Those types of transmissions usually have a foward and reverse clutch pack and one for each speed. So for any speed you need to operate one of the direction clutches and one of the speed clutches and all the others are off. I once found a piece of rubber from the inside of a hose in the valve body. That caused erratic opertion and that was really hard to find.
It sound like yours does not fully go into gear, so that is why I think it is a control issue.
When it is slugish does the engine rev freely and the machine not move or is the engine loaded and the machine not move?