Just to clarify some terminology. The "clutch" on your mower is just a pedal/lever/pulley system that makes the drive belt loose or tight by using
the foot control. Your gears are working, you are just having trouble shifting while the engine is running.
Your problem is, that your belt keeps turning the transmission pulley even when the clutch pedal is fully depressed. Normally this is caused by having the wrong size belt on there, but a dragging idler pulley, or misrouted belt could be another cause, but I think this is your trouble area.