99 Silverado problem when taking off. Would it be rear end, Tranny, Etc??? Help!!!

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I have a 99 Silverado 3/4 ton 4x4 Extended Cab, Short bed with the 6.0 liter engine and automatic transmission. The truck has approximately 125k miles on it. I bought it used with a 117k. I bought this truck back in February of this year and immediately starting towing a 24' Enclosed Race Car Trailer with it.

The problem I am having with it is when I take off from a dead stop I have some play somewhere in the rear end, tranny, etc. The pickup will actually take off and start to move and then it feels like something frees up and there is some play somewhere in the drive line just for a second, but you notice it after the truck starts to move if that makes sense. When I first started noticing it, I thought that the ball or the pin on my reciever hitch was loose and when I would take off there was some slop in it before it would start to move the trailer. But after unhooking the trailer it does the same thing. The transmission and everything feels real solid when driving, towing, and shifting.

Has anyone had any trouble like this? I am not sure but was wondering if a limited slip differential would act like this after taking off. Any help would be greatly appreciated. Thanks a bunch!!!!