There are variose types of LSD Toyota use a clutch pack type that basicly lightly locks the two sides together under spring tension and locks up harder and harder the more touque is put threw the diff.So when one wheel is lightly loaded or in the air you cant put much if any tourqe to lock it up. a viscous or cone type would be better in this situation as lock up and tourqe transfer to the other wheel is speed differnce dependant.
So to try and over come this .Stop and take off with the hand brake applied firmly,the extra tourqe required to turn the wheels force the side gears in the nugget out against the clutch packs locking the diff up tighter(but not actually locking it in a mecanical sense)
Jack both rear wheels up off the ground in gear hand brake off. Try to spin a wheel if it turns easily and the oppisit turns the other way. You either have a plain old open diff or a very very worn lsd.if its hard to turn for a start then loosens up to easy you have a worn lsd.If its very difficult start to turn then eases up to a bit of resistance its a good one.
Lots of LSDs perform poorly or not at all because they have had regular diff oil put in them.If with the obove checks show you do indeed have a LSD change the oil with proper LSD oil and do it again a week or two later. This will certainly improve it even if it is worn.
Otherwise there is one under my bench I could shim up tight as fuc for ya and you would just have to get someone to fit the ringgear and set backlash for ya if your not confident doing it yourself.
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