Correctly setting up a ring & pinion takes specialized tools & experience, it's also a "get-it-right-the-first-time" deal - if you don't, by the time you know it's not right, you've already destroyed the gears.
Lockright would like you to think that anyone can install their locker, but, because you have to remove the differential case from the third member, there's always a possibility that you don't get the mesh correct on reassembly.
The loud whine of the first differential sounds like the gears may have been incorrectly set up, there's insufficient information about the second failure to make any suggestions as to what failed - you'll need to at least remove the third member to know whether a rebuild is even possible, and that's also the point at which you'll know if the failure relates to your installation of the lockright - maybe you got it right and something else failed.
Technically speaking, when you install a lockright, you have to remove the differential - once the gears come out, there's no longer any differential action, so no, the lockright is not hard on diffs, but, lockers generally load the axles to a greater extent, and depending on the terrain you're driving, you can end up applying 100% of the available torque through a single axle shaft, and that becomes the point of failure.