Forget the JSON, CSV, XLS, Google Drive, Etc..
Here is how to export-import board from one account to another account without losing cards or card images: (You only lose comments)
Copy Name of the board to paste to the new copied board
BOARD TO COPY => Show menu - more - copy board - PASTE: Name you copied above or any name - TEAM: none - KEEP CARDS - GO!...
On mobile app you can use move card functionality.
Click card and then three dots on the top right corner. From there choose "move card". This opens a dialog where you can enter a new position.
This way much easier than dragging if the list is long. Hope this helps!
As of July 2019, the above steps didn't quite work for me. What I had to do was:
Hover over the sticker
Enter edit mode (press "e" on the keyboard)
Left-Click the sticker
Click "Remove" on the pop-up menus
Simply hovering and clicking the "Remove" appears to be ignored.