Let's say you have a structure $A$ of some language and a set of all sentences that are true in such structure $Th(A)$ of the same language.
Then $Th(A)$ is supposedly a complete theory. I fully believe that $A$ is a model of $Th(A)$ but have hard time understanding why should it be the only one.
I mean, why can't we have a structure $B$ in which all $Th(A)$ sentences are true but where one additional sentence is true. Then it would still be a model of $Th(A)$ (as $Th(A)$ is subset of $Th(B)$) which means $Th(A)$ wouldn't be a complete theory (it would have two non-equivalent models).
Let $\varphi$ be any sentence of the language $L$, and let $A$ be an $L$-structure. Then one of $\varphi$ or $\lnot\varphi$ is true in $A$. Hence $\text{Th}(A)$ is complete.
We look in particular at your structure $B$. Let $\psi$ be the "additional" sentence of $L$ which is true in $B$. If $\psi$ was not true in $A$, then $\lnot\psi$ was true in $A$, so is an axiom of $\text{Th}(A)$. Since $B$ is a model of $\text{Th}(A)$, it follows that $\psi$ cannot be true in $B$.
It is perfectly possible by extending $L$ to a new language $L'$, to produce a new sentence $\psi$ and an $L'$-structure $B$ in which $\psi$ is true. But this can only be done with a sentence that is not a sentence of $L$.