Check key for shear on axles.
With a helper sitting on the mower. Remove plastic cover off center of the rear wheels. Have helper try and go forward or reverse. If the axle(center shaft) rotates but the wheel doesn't, remove wheel to find out why.
Check key for shear on the pulley with fan blade on top of the transmission. You might be able to lift up the seat to inspect it. It should be obvious without runnnig the engine if the pulley is loose.
I had one and it was jumpy like that. I could see how it would shear one of those woodruff keys.