If the prop spins, but the shaft doesn't, that sounds like a spun prop hub..
(The rubber bushing between the center of the prop, and the blades)
You say it vibrates on acceleration.. Does it get up to speed or just rev and go nowhere.. Like you're on ice?
Anyway,
Not a pro here, but my bet is spun hub, but I wouldn't rule out bent shaft either....
How did you check if propshaft was true, if the blades move but shaft doesn't?
(the prop and shaft should have splines which interlock, so prop spinning, shaft not, is not because of a loose nut.)
Hope it helps!