Did you register a full AdobeID with ADE on the previous computer?
If so, register the same ID on the new computer and things should be fine.
Sometimes ADE gets its registration/activation confused and in a semi-authorized state.
Just in case, try completely removing any authorization using ctrl-shift-D to ADE (cmd-shift-D if on Mac).
Restart ADE, and then reauthorize with your Adobe ID.
If you did not use a full ID on the previous computer, but just 'authorize without ID', you may be stuck.
This is a vicious trap that Adobe lays for unsuspecting customers.
When you register 'without ID', ADE creates an anonymous/implicit ID with limited powers (eg can't be used to share with other devices).
Whenever you first load a DRM book, that copy of the book is associated with whatever ID the computer is registered to at the time.
Any book you load while this anonymous ID is active gets associated with that ID, and can't be read on any other device.
When you properly register your computer with a real ID, that old anonymous ID is lost.
Now you don't have any devices that can read the book.