Here's my 2 cents... Posi (Limited Slip) should not be causing your problem, cuz your going straight (assuming the rolling diameters of both rear wheels is the same or pretty close). The clutches in LS only come into play if there is a greatly differing load or rotation speed wheel-to-wheel. If something was broke big time, would think you would have a no-go situation. When the clutches eventually wear out, you will be effectively left with a regular "open" differential (not LS).
Some ideas... have you checked your leaf springs closely? Maybe u-bolts loose/broken, leaf or two broken (look at edges real carefully with a light, looking for a vertical crack, have seen broken leaves that looked OK when parked!). What about your lift pieces - mountings, brackets, etc. What I'm getting at here is anything that allows the rear axle to move fore-and-aft on one side will translate to an unwanted steering input.
Probably want to look at the front end also. When you get on it, weight transfers from front to back, so front suspension unloads some and the body rises. Anything there that could give a steering input? I'm assuming that when this happens, that your transfer case is in 2WD, and that the front shaft is really not powered, and that BOTH hubs are really unlocked and stay unlocked. Any funny business there could create a pull.
Too bad you didn't drive it a lot BEFORE the lift was installed, that way you would have a good reference. Oh well!