Dropbox support got back to me with this great response.
If you want to make permanent changes to your computer that do not sync back to the website you need to unlink the computer first.
Follow these steps to unlink a computer:
- Click the Dropbox icon in the menu bar at the top of your screen on Mac.
- Click on the gear icon in the Notifications panel and select 'Preferences' > 'Account'
- Click the 'Unlink' button
- Afterward, Dropbox will prompt you to re-register. Quit!
You can now safely move or delete files in the Dropbox folder and not affect anything else.
Note that the Dropbox folder itself is still there. If you relink to a different Dropbox account, the application will detect that another Dropbox folder already exists on the computer, and will give you two options: 'Choose Another Location' or 'Install'.
If you want your new Dropbox folder to be in a location different than your existing folder, click on 'Choose Another Location'.
If you want your new Dropbox folder to be in the same location as your existing one, click on 'Install'. This will rename the existing Dropbox folder to 'Dropbox (Old)' and will create a new Dropbox folder for the account you’re currently linking.
The above steps should give you the option to keep the files in an unlinked folder (Dropbox Old) or you can choose to delete that folder.
You may also want to unlink, delete the folder entirely, reinstall the Dropbox application and link the device. This will then create a new Dropbox folder and the functionality of the Dropbox application is limited to watching your Dropbox folder for changes and syncing any changes that occur across your connected devices, and will sync down your Dropbox account that shows on Dropbox.com to that device.