Thank you, Painted Lady. This pointed me in the right direction, even though my path wound up being a little different. On my Mac, I had to use iTunes instead of the Finder. This is what wound up working:
Connect old iPad to Mac.
Open iTunes, connect to old iPad, click on its "File Sharing" option.
Find the Queen's Wish app and create a local copy of the saved files.
Connect new iPad too Mac.
In iTunes, connect to new iPad, and click on its "File Sharing option.
Find the Queen's Wish app and copy the saved files, replacing the default ones.
I've confirmed that I can open the saves on my new iPad. Thank you for saving me from a lot of backtracking!