Should I be pruning the dead parts of my string of pearls...

6β pot with drainage

Last watered 4 weeks ago
It looks slightly over watered @PlantDADHD. I would suggest repotting to a chunky/gritty succulent mix with added perlite for extra drainage and aeration in a terracotta pot. Also, only water when the soil is COMPLETELY dry.
I always recommend worm castings ( a tbs or two) in with the substrate when repotting. I did it with my string of pearls I got a few weeks ago and theyβre booming with little babies and new little trails. And my wife brought it home because it was dying at the store and was also browing much like yours.