2005 F350 superduty transmission

Ok guys, question. I was hauling a bobcat and I tried to reverse and my truck would move. It would go in first but not anyother gear.

I took it to the ford dealership and they told me that the reverse gear was cracked. Also, the turbo needs to get sycronized.they recomended to replace the entire transmission. Do you agree? They said this would cost 3900$. Is this a good cost? Do you have any recomendations on if dealerships are better than taking it to a (local guy)? Thanks

Is this transmission an automatic or a manual? When you say “I tried reverse and my truck would move”, did you mean “not move”? Was there any noise associated with this attempt to go in reverse?

I have never heard of syncornizing a turbo so could you elaborate on that a bit.

Hope to help.