Welcome to the forum.
Dropbox is a sync service. As such, it stores copies (for sync) of whatever is in the source folder for Dropbox. When you say "move" are you implying that you no longer want these docs and folders on your Mac, but just at Dropbox? I don't think that can be done.
The home folder is key to how the operating system works, so you cannot just do away with it altogether, as you found out.
Apple provide a service in iCloud whereby you can store your home folder documents in iCloud and not have them on the Mac until you call for them. Is that what you are trying to do?
Finally, what Mac are we talking about?