You want a locking differential, for sure.
Limited slip still allows the wheel with the least traction to spin, and only puts some of the power to the other wheel. Sometimes it's enough, sometimes it's not.
The Eaton locker will not lock up at speeds over 25MPH or 45MPH, not sure which, and will unlock automatically at that speed. So don't just mash the pedal and try to spin your way out.
On slick surfaces, having both wheels spinning can be a blessing and a curse. You will get more fishtailing of the truck due to the fact that there's no stationary wheel to "anchor" the rear end from sliding around.