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I was wondering if all expys have limited slip or not. I'm pretty sure I don't because when my tires spin, I only have one front and back that spin. What would I need to do to get limited slip, are they easy to find in junk yards?
Unfortunately what you are experiencing is normal, even if you have a limited slip rear differential. The lsd will try to slow a spinning wheel by applying pressure to the clutch pack in the diff that send power back to the other side. It slips enough though that one tire on ice/snow and one on pavement is going to result in spinning even with the lsd. Differential lockers are a different story but they can lead to instability at speeds over about 20 mph.
Yes you could find one in the junkyard but it would probably be worn out. There are aftermarket versions that are much more effective (tighter clutches) but by the time you buy the new diff and install it you have spent $5oo to DIY, or double to triple that set up by a mechanic.
My Navigator 4AWD has a LSD rear differential and my experience is that the rear wheels will lock up and when the A4WD module notices a speed difference from front to rear through the speed sensors it engages the front axle which does not have a LSD and that can and often does result in one front wheel engageing and the other not. There are many types of LSDs as well as Lockers and they pretty much all operate differently and will give the novice a bit of a fright when they engage unexpectedly in the wet or on ice. We get a good bit of "Black Ice" in my area and for that reason I am going to install a toggle switch in the system to disengage the A4WD system when I want 2WD only.
Years ago I drove in Chicago winters with a "Lincoln Locker" in a heavy car and had no problems with control, a A4WD or 4WD with a locker is a whole different animal though.