If a differental pin is broke and it is stuck in first gear is it possible to drive with transmission fluid in?

My van started to leak tranmission fluid so we put fluid in it and stop leak. Which both leaked out.But when you put tranmission fluid in it the van runs normal until the fluid leaks out. The dealership is telling me now that the differental pin is broken and it is stuck in first gear so I need a whole new tranmission. But when they towed it the van was in nuetral and not stuck.I am wondering if what they are telling me is possiable. Its a 97 plymouth voyager wagon.