There is no way to transfer the collections directly or even via computer.
You could move the books from the directories folder of Old Kindle to a computer, and then copy them into the New Kindle, or use Calibre. That part would be fast and easy. Creating the "Collections" and populating them will be a strictly manual process that will have to be done on the New Kindle. I have helped several friends and family members set up new Kindles, and this seems to be the only way to recreate on the new Kindle the setup you describe.
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