Why am I having problems shifting my garden tractor?

Why am I able to manually push my riding mower while it is in any gear. It rolls freely. It has a peerless transaxle--model 206-545C. The vehicle always jerked while in low gear. Lately it will not shift to go. All belts, linkages, pulleys are operable and tight. I have it jacked up with the wheels off and the transmission is visible after detaching the rear shield. I am able to start the engine, watch the pulley turn properly. I can manually turn the axle with my hand while the shift lever is in gear. I believe that this can only be done while the unit is in neutral. This all started when I became stuck and I tried rocking the unit back and forth and shifting from gear to gear.
What have I screwed up and how can I possibly fix it?