I bet the books did not carry over because this would amount to the creation of another copy of the book and I bet that's something DRM won't allow in this case.
In the Overdrive app on the new iPad, go to "Get Books" and open up your library's overdrive page. Click on "My Account" then on "My Checkouts." Assuming you have only downloaded an item once, you should be able to click "download" again. I think you can download something only twice (maybe three times? Not sure) during your checkout period. If you have reached the download limit, contact your local library because they should be able to reset the download limit provided that the book has not expired. You will not have your bookmarks preserved, etc, but you should be able to get your books.
Signed, your friendly neighborhood librarian.