I think it is highly likely that the hydrauoic linkage is not fully releasing the clutch even though the pedal is fully depressed.
Try CHANGING the clutch fluid (and the brake fluid while you are at it). With new fluid, make sure it is thoroughly bled.
It is a clutch problem of some sort. The key here is that it shifts good with the engine off. For some reason the clutch is dragging. I'm betting that clean clutch fluid with no air in the system will solve the problem.
