Aldiko is one of my favorite apps and one of the older versions I keep today (from Android 2.3 no less). Any downloaded book is kept in the Ebooks folder under a subfolder with the titles. So copy and paste all those subfolders to the Ebooks folder in your new device and re-open the Aldiko app and they should populate the bookshelf. You can also read PDF files there too, as using Aldiko to open a PDF (such as a service manual in my case) makes them also appear on the bookshelf.
I don't use paid content or an account with them though, I just use it for free content. So I am not sure what paid content does or where it's stored, but if you use an Aldiko account it should just work like above once you're logged in.