Hello Mike,
this is NOT supposed to happen, because Notebooks tries to synchronize all moves and deletes, too.
- Are you using any special characters in the titles of the books that might confuse Dropbox?
- Did you receive any error messages?
- What happens when you delete the duplicate note in Notebooks, is it deleted on DB as well?
If this problem persists, I would recommend to reset the Dropbox sync history (in Notebook's settings) and start from scratch. The first sync will then report a conflict for all documents that are in NB and already on DB, but this is ok and you can ignore that. After that, consistency should be reestablished.
I hope this helps.
Best,
Alfons