In particular, in view of the focus of your studies, I suggest the following additional book; where additional is meant that I would not suggest it as the only book (see below for explanation).
There is a fairly recent book (in two volumes) by Henri Cohen entitled "Number Theory" (Graduate Texts in Mathematics, Volumes 239 and 240, Springer).
[To avoid any risk of confusion: these are not the two GTM-books by the same author on computational number theory.]
It contains material related to Diophantine equations and the tools used to study them, in particular, but not only, those from Algebraic Number Theory.
Yet, this is not really an introduction to Algebraic Number Theory; while the book contains a chapter Basic Algebraic Number Theory, covering the 'standard results', it does not contain all proofs and the author explictly refers to other books (including several of those already mentioned).
However, I could imagine that a rich exposition of how the theory you are learning can be applied to various Diophantine problems could be valuable.
Final note: the book is in two volumes, the second one is mainlyon analytic tools, linear forms in logarithms and modular forms applied to Diophantine equations; for the present context (or at least initially), the first volume is the relevant one.
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