So your linked files are at a *single-folder* path on Cloudme and you wish to move them to a single-folder path on Dropbox ?

If so, the best approach would have been to set a Linked Attachment Base Directory (LABD) setting in Zotero settings, before moving any files in your OS. That simply tells Zotero to use that LABD path as the parent path, when trying to open existing linked PDFs. Then after you move the files in your OS from Cloudme to Dropbox, you would just update the LABD to the new location.

The LABD is most often used where two computers have linked files at different parent locations (eg D:\Dropbox on one and E:\Dropbox on the other). But it also works for any situation where linked files now have a different parent path on a single computer.
https://www.zotero.org/support/preferences/advanced#files_and_folders

So if by chance you *already* had a LABD setting in place before the move, all you have to do now is update it.

If not, in Zotero when you try to open a linked file (now at Dropbox location) that is not where Zotero thinks it should be (at its original location at Cloudme), the code around LABDs is also able to recover from (some of) those situations, and to apply that fix more broadly. See 'Automatic Linked-File Relinking' under the LABD section:
https://www.zotero.org/support/preferences/advanced#files_and_folders

Zotmoov is used for moving 'stored' files to linked files (and it has to be able to find the stored file under Zotero\storage in order to do that). You do say you have a mix of stored and linked files. So if by "I want to transfer all attachments to a Dropbox folder" you now want *all* you files to be linked files, you would set Zotmoov's 'move to' directory to your linked-files directory, and then use right-click Zotmoov: Move Selected to Directory to move the stored files. And if you wanted all *future* files downloaded by the Zotero connector to be converted to linked files, you would turn on 'Automatically move/copy files when added'.

As with all Zotero use, but especially before any significant changes, make sure you have a backup of your Zotero data directory.
https://www.zotero.org/support/zotero_data#backing_up_your_zotero_data

Zotmoov's Move Selected to Directory can be run on as many or as few *selected* items as you wish ... and in fact with lots of items it is best done in small chunks (eg all the 'A' titles, then all the 'B's', etc). So you can also test that it is doing what you want on a small chunk of items first. You would run the Zotmoov move operation at the Library level in Zotero, so that you cover all items.

It sounds like your stored files are on WebDAV ('prop + zip pairs' ... WebDAV only works with stored files). I am not 100% sure, but you may be able to have Zotmoov pull PDFs directly from WebDAV to your 'move to' linked files folder as actual PDFs (ie without having to turn off WebDAV and revert to standard stored PDFs first). This suggests that you can do just that ...
https://forums.zotero.org/discussion/75470/move-from-zotero-storage-synced-over-webdav-to-syncd-folder-based-linked-files
(in that case using the older Zotfile, the predecessor of Zotmoov)

If file renaming is set to ON in Zotero settings, the PDF filename should be renamed when Zotmoov moves it to your linked files folder. But if your renaming scheme is unchanged, the name won't change.

BTW if you want to maintain access to your metadata on the web library and other devices, keep Zotero *data syncing* turned on (data sycing is free and unlimited). But file syncing would be off (file syncing doesn't work with linked files).
https://www.zotero.org/support/attaching_files#stored_files_and_linked_files

Zotmoov just moves the stored PDF from its (arbitrarily-named) folder under Zotero\storage to your linked-files ('move to') folder. But your never deal with that source folder - you just select items in Zotero's Library view and all that is handled 'under the hood'. The only caveat is that Zotmoov must be able to find the PDF in its stored folder (where the Zotero database says it is located) in order to move it. Sometimes people have moved things around outside Zotero that messes that up. But if Zotero can open a PDF, that shows it is where it should be, and will be movable by Zotmoov.

As I suggested, try it out by selecting an item in your Library that you know has a stored PDF (no chain link on the PDF icon that signifies it's already a linked PDF).

You don't have to do that (and there is no easy way to select just items with stored attachments anyway AFAIK*). Just select all the items in a chunk and Zotmoov will only move the stored files (and also any existing linked files which are in a folder *other than* your current Zotmoov's 'move to' folder), skipping the already-linked files already in the 'move to' folder.

If you have Zotero's file renaming turned on, both the stored files *and* all existing linked files in your selection will be renamed (even though the latter are not being moved). So if you have any existing file names that you don't want to have changed (to your current naming scheme), turn renaming off.

*you can get a list of all attachment file paths with Zutilo, or a list of all stored or linked files with javascript.

BTW I notice that Zotmoov no longer attaches the pdf to the Library item generated when the pdf is dropped as a standalone item into the Library. One then has to drop the pdf onto the newly generated Library item as a linked file attachemnt. Is this a bug?