When you first had ADE did you register it with an Adobe ID?
If so, register the new ADE with the same Adobe ID.
If not, I am afraid you may be scuppered ... see below.
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.
ADE probably gave very you a mild warning that this could happen when you registered 'without ID', but not nearly strong enough.