Would a 2003 AWD Explorer benefit from a limited slip rear differential?

I bought a used 2003 AWD Explorer 4 door that has the 3.55 open differential rear axle. A salesman told me that Ford's AWD sends power to all four wheels thus I do not need to have a limited slip diff. I'm skeptical of this but have not found a good source of information. My guess is that AWD sends power to each axle, not each wheel and that an open diff rear axle would behave the same on a 2, 4 or all wheel drive system. If this is true then I'm going to have a shop add the LS carrier to the 3.55 gears. I live in snow country, not too much ice, and occasionally drive on unimproved roads. I have had good luck with LS rear diffs. A shop in Reno, NV will install either the Ford or the Auburn carrier for about $600 with the Ford costing about $75 more. They say that the Auburn carrier is a better unit. So, does AWD obviate the need for LS? If not, Ford or Auburn? Thanks, Chuck