Just fishing to see if it is just me or they are all close in this aspect.
No, it is not usual with a LSD. You should no problem accelerating in a turn on dry ground. Both tires should plant and spin at the same time once the LSD activates. On dry ground you will always skid the inside tire becuase there is no bias and both wheels try to turn the same speed.
Does this happen worse one direction or is the same turning left or right?
There have been a few that reported this same result with an LSD. Some never got it to work, others got in a tight turn under power and load and heard a loud bang from the rear end and the LSD starting working perfectly after that.
If you can't turn the tire, put the transmission in neutral and try again. If both tires spin the same direction, the limited slip works, although maybe not fully. If only one tire turns and the driveshaft turns, the limited slip is broken or you don't have one.
Nope, that is not the way a Tq bias LSD works. On these trucks it will act just like an open diff spinning the wheels. The LSD will only activate with input TQ on the pinion and if the wheels speed is different from side to side. Otherwise you never know its there.
The Torenson type LSD consists of a helical cone inside the spider gear of each side. When the wheel speed and axle speed are different the helical gear walks a brake out to the spider and brakes the faster spinning axle to transfer TQ to the slow one. Once the speed equalizes the brake cones withdraw to static position.
There are six small brake shoes on the brake cone that apply the needed friction to slow the over speed axle. Several problems can occur with this type of setup. If you hammer the throttle too hard with one wheel spinning the brakes cannot handle the TQ and will eventually burn the friction material off. Also, if the wrong fludi is used or over use occurs the frictions will glaze and not work either. The other bigger problem is the helical cone will jam in its receptacle and not activate correctly. All it takes is a sliver of metal from casting or a rough edge in the mating and it locks itself up. Possibly it could be due to lack of use that the seize also.
Either way, if you actually have an LS and you one wheel burn all the time the diff needs disassembled and the problem found if you want to work. When the LS works correctly on a slightly slick road in a turn a heavy bump of the throttle will swing the rear end out of the turn with little spin and line it up straight. You have to get on and off the throttle to stop the rotation but once you get the feel its hit the throttle lift, wait a count, and nail it again and you launch straight.
If the LS doesn't work the truck just rolls thru the turn with one tire spinning and little acceleration while you wonder what anchor has been deployed. Another thing to try is adding a little brake if the spin starts. This will help slow the over speed side and get the LS to activate and hold better given it is functional.
Attention: TDR Forum Junkies 
. Dealer pulled spec sheet and if it is correct I have a LSD also they opened her up and "found nothing". I have removed and installed LSDs of similar design,(along with lockers and plenty of gear sets) don't remember is this the kind that looks like a normal open diff or has a "solid" case around the moving parts i. e. Detroit lockers. I have changed the fluid myself but that was a while ago and the old timers has set in. 
