I ran the build codes (sticker in glove box) on my '96 T-10 Blazer and one shows a limited slip rear. If I spin the back tires only the right rear spins. Is there anything I can do without disassembly to try to coax it to work?

 

It all depends on how much traction you have side to side. Even a posi unit may let one tire spin while the other doesn't, but I would NOT do burn outs if you are concerned about your diff.

The limited slip diffs used in the S-series trucks is a friction type posi with a locking cam for slow speed slips... Here's the description:
If you have been doing burnouts, it is possible that you have sheared off the locking cam by doing so. Burn outs are not healthy for any vehicle and can seriously screw things up without proper attention.

 

In all honesty, I really don't make a habit of doing 'burnouts'. I was using that as a way of describing the way one tire will turn and the other receives no power to speak of.

If i'm in wet grass or mud and I gently ease into the throttle but lose traction, only the right wheel seems to receive any amount of power, or if it has traction, the left will slip easily.

I have had limited slip before and i feel like something is not working properly with this one.