The obvious is a bad universal joint or an unbalanced drive shaft causing the vibration. If you think it could be the tires, is it possible to swap the front and rear tires, or do you have bigger tires in the rear?
As for the late shifts, do you know if the tranny has a shift kit installed? These late shifts could be intentional if so, but as Chryco mentioned, a shortening of the kickdown linkage would soften it up. The linkage is adjustable.