LSD not working correctly

[tucker0104]

I have a 97 f-250 4x4 with a C5 or C9 differential code which I am pretty sure means I have a LSD. I don't think the LSD is working correctly. I pull my trailer out of the water on the boat ramp and as soon as one rear tire gets in the grass it will spin like crazy and the other tire just sits there and does nothing. From what I understand about LSD, the other tire should engage and help the other tire out. What coule be the reason for it not engaging? I changed the diff oil out with Amsoil 75W140 a while ago. Thanks for the help.