When you say "it" says you have to register with ADE, what is "it"? A manual? An error message on the screen? Or just a general note to everyone on your library website? When you created the Sony bookstore account, then it should have also created an Adobe ID for you with the same email address and password.
You can tell if your T1 is authorized by this step. Click on "Settings" on the second page of the homescreen. Then click About and then click Device Information. You should see "This device is authorized for reading and purchasing Adobe DRM content".