You may have a corrupted address book(s), but first I suggest you create 2 address books (File/New Address Book...), test1 and test2, add a new generic contact to each one, and then see if you can move (drag) or copy (Ctrl-drag) contacts between the two address books.

False (ThunderBird) Alarm: This turned out to be a Windows (7) problem, Apparently Windows Drag & Drop will stop functioning (for reasons not clear to me yet), ReBooting resolves the issue. There are other suggestions to resolve which I have not been able to vet. I discovered it was not a ThunderBird when I realized Drag & Drop was also not functioning in Windows Explorer.

r