I can use settings/"check now" and "scan now" to scan for new songs.  But how do I scan for new *cover art*?  I just want to keep exactly the same songs I have now and look only for changes in cover art associated with those songs.

You can do that by selecting 2 or more albums in the album list (ctrl-a to select all) and select auto-select pictures, it will search for a picture for the selected albums that don't have a valid cover.