Your concern does not sound like a normal limited-slip differential (LSD) concern. Chevy calls theirs Posi-Traction. Basically it is a clutch pack that is splined to the axle shafts, and effectively locks the shafts together when driving straight, and unlocks when one wheel is going faster than the other (turns).
When the clutches wear out, the axle shafts are spinning independent of each other, and the LSD is no longer working. This causes all power to transfer to 1 wheel only, same as an open differential.
Your concern is the opposite of that. Did this just recently happen? Or has it been like this since you bought your truck?
I'd pop off the rear axle cover and take a gander at the LSD, see WTF is going on. Very simple mechanical system, you can rebuild it yourself in a couple hours if you are so inclined.
1990 Mustang LX 5.0 - Five oh!
1994 S10 Blazer - Fix n Flipper
2005 Ranger 4.0 - Daily Driver/Tow vehicle