Clutch
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It not going into gear could be a variety of things. Worn selector forks, bearings, clutch assembly, clutch springs.
Sounds to me like the clutch isn't disengaging.
Again what I said earlier will tell you it is clutch disengagement problem.
Never heard of a worn basket on a dt before though. Maybe if it ran with no oil. My brother ran his dt gearbox on coolant and oil with no problems. So they are pretty well engineered.