6th gear is synchronized, but if you shift too fast, you can beat the synchro's balk ring and grind the synchro teeth due to the high rotating mass of the gears, clutch disk, etc. Once in a great while I notice a little grind going into 3rd, even at 16,700 miles, but I've never noticed it going into 6th. Of course, I don't try to shift the NV5600 like it's a Ford T&C top loader with a Hurst Competition Plus, either. If you bought the truck used, the previous owner might have the wrong lubricant in the transmission as well.
On the LSD, the "tightness" of these things vary all over the map from truck to truck. If it's not worn out (you don't mention how many miles are on your truck), it may have too much friction modifier in the rear axle lubricant - that means it won't chatter on a turn, but it won't function properly, either.
Rusty