I am also experiencing this issue, also with Namecheap as my hosting provider. I also see the two menu items as described by the OP. I noticed that the timestamps on my /wp-content/litespeed subdirectory and its contents tend to reflect times from today when I was using Litespeed, while timestamps for the /wp-content/plugins/litespeed-cache subdirectory and its contents are from a couple of days ago (maybe when I last updated the plugin).
