Hello DocumentGeek,
As per the description above, you want to move PDF files into a folder on your iPad, Is that correct?
Please refer to the following KB doc which discusses how to manage your Document cloud files on an iPad Acrobat Reader Help for iOS: Managing files and folders Link: https://www.adobe.com/devnet-docs/rm/ios/acrobatios_en/manage_files.html#movefiles
If you wish to use the "Organize Pages" tool, for this, you need a subscription to "Adobe Acrobat Pro" or "Creative Cloud" which includes Acrobat Pro with your Adobe ID.
Acrobat Reader Help for iOS: Organize pages Link: https://www.adobe.com/devnet-docs/rm/ios/acrobatios_en/organize.html
Adobe Acrobat Reader for iPhone and iPad displays the 'Organize Pages' option in the Tool Switcher, only if you are subscribed to the "Acrobat Acrobat Pro DC" subscription service and signed in.
Feel free to update this discussion for any further assistance.
Regards,
Anand Sri.
[Edited response]