In the question as asked, your mistake is the following:
if the logic has two non-isomorphic models, then this is rather trivial, or? For then just choose a proposition which is true in one model, but false in the other...
Just because $\mathcal{A}\not\cong\mathcal{B}$ does not mean that there is a sentence true in one and false in the other! There are non-isomorphic structures which are elementarily equivalent. One easy way to see this is just by a cardinality argument: in a given language $L$, there are only $2^{\vert L\vert+{\aleph_0}}$-many sets of sentences in the language $L$ but a proper class of structures, so we have to be able to find non-isomorphic structures with the same theory. More satisfyingly, the compactness theorem (or rather, its corollary the upwards Lowenheim-Skolem theorem) tells us that for any structure $\mathcal{A}$ there is a nonisomorphic, elementarily equivalent structure $\mathcal{B}$. And in particular, if $T$ is a complete theory with an infinite model, then $T$ has lots of nonisomorphic but elementarily equivalent models (since any two models of $T$ are elementarily equivalent since $T$ is complete). (Exercise: "has non-elementarily-equivalent models" is exactly the same as "is incomplete.")
So Godel's incompleteness theorem says far more than just that there are non-isomorphic models of any "reasonable" theory of arithmetic: it says that no reasonable theory of arithmetic even pins down the true natural numbers up to elementary equivalence!
It's worth considering, at this point, some examples of complete theories.
The theory of dense linear orders without endpoints is computably (indeed, finitely) axiomatizable and is complete; this is a standard exercise in most first-semester logic classes.
A more interesting example is the following. If we consider the structure $(\mathbb{N}; +)$, it turns out that Godel fails: the theory Presburger arithmetic is complete (it axiomatizes $Th(\mathbb{N}; +)$). It's also decidable, since Presburger arithmetic is computably axiomatized.
Similarly, it turns out that the theory of real closed fields is complete (and computably axiomatized); this is due to Tarski.
You also asked about truth (although truth doesn't even show up in your statement of Godel's theorem). The issue here - as in your other question - is that you are conflating two notions of truth. The truthy version of Godel's incompleteness theorem says
For any "reasonable" theory $T$, there is a true sentence $\varphi$ which is not provable in $T$.
"True" here means "true in the structure $(\mathbb{N}; +, \times)$;" this has nothing to do with truth in all models of $T$. A better statement of the "truthy" incompleteness theory is:
For any "reasonable" theory $T$, there is a sentence $\varphi$ such that $\varphi$ is not provable in $T$ but $(\mathbb{N}; +,\times)\models\varphi$.
This makes it very clear what's going on, and why Godel's completeness theorem is irrelevant. Again, it boils down to the fact that any reasonable theory of arithmetic $T$ will have lots of non-elementarily-equivalent models; the sentence $\varphi$ above is not true in all of them, hence (by Completeness) not provable in $T$, but is true in the standard model.
(Keep in mind that "reasonable" here means, among other things, "computably axiomatizable"; of course the set of all sentences true in $(\mathbb{N}; +,\times)$ is a complete theory, but Incompleteness doesn't apply to it since it's not computably axiomatizable.)
