How can I copy a Playlist from one iTunes library to another?
Under the File menu, In the Library expanding list, you will find "Export Playlist".

On the receiving machine you would use "Import Playlist" found in the same menu.

Seems like you have the right procedure (although I do not know for sure that drag and drop will work with the playlist...) but have you tried to export a .txt file - that's what I get when I export a playlist. If you have the option could you export it as txt, verify that the file contains a list of songs, then verify that it also contains a list of songs when you get it back from dropbox? You should at least be able to isolate where the list disappears.