What I do in this situation is to delete them and then reinsert to the correct category(s).
Download the ones to be moved and change the status to 9 and upload. This will delete the product and the image. If you change the image folder name first, then the images should be left untouched.
Change the target folders and the status back to 1 (and rename the image folder back to what it was) and you will be done.
I would try a test product first and be sure to backup your db.
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