I did not study scattering theory much, but here is what I think:
It is always good to confront more than one good book. The books you mentioned are standard well-known textbooks, so every one is definitely worth looking into.
The book on quantum theory by Landau and Lifgarbagez is a very good one, particularly because of the very interesting introductory chapters where they explain also subtler points concerning Schroedinger equation and its solutions, density matrix etc.
However, when I came to scattering theory, L&L was too hard for me, I tried for few days but I did not get through. It requires quite a lot of math and is quite terse for a beginner. Sakurai may be better for first reading.
Also, this book may be of help:
N. F. Mott, H. S. W. Massey, Theory of Atomic Collisions, Oxford University Press, 1987