I elaborate my previous question about example of complete theory, that has two models, $\mathfrak{A}$ and $\mathfrak{B}$, that $\mathfrak{A} \le_{end} \mathfrak{B}$, but $\mathfrak{A} \npreceq \mathfrak{B}$.
Could you prompt me, now, such example with predicate symbol in its signature? Or, maybe with function. Cause I need to do some task(I will do it myself:), about checking of keeping truth over end extensions for non sigma - formulas, so I need some theories, where I would check it for formulas with $\forall$ quantifiers, in way, it would be more convenient.
Update: And, if it possible, can it be theories on some intervals, like (0,3) or something like that? Initially I wanted to take theory of dense linear order without endpoints over field of rationals with only multiplication and without addition, but I ain't sure, that it will be right.
The answer to your original question ($\mathfrak{B}=$ $\mathbb{N}$ with the reversed order as an end extension of $\mathfrak{A}_m=\mathbb{N}_{>m}$ with the reversed order, for your favorite $m\in\mathbb{N}$) can be modified in various ways. Specifically, say that a $k$-ary relation $R$ on $\mathbb{N}$ is $m$-shift invariant iff we have $R(a_1+m,...,a_k+m)\iff R(a_1,...,a_k)$, and say that a $k$-ary function $f$ on $\mathbb{N}$ is $m$-shift invariant iff we have $f(a_1+m,..., a_k+m)=f(a_1,...,a_k)+m$. Given any $m$-shift invariant relations and functions, we have that the expansion of $\mathfrak{B}$ by those relations and functions is an elementary equivalent end extension of the expansion of $\mathfrak{A}_m$ by (the restrictions of) those same relations and functions, while of course this extension is not elementary. (Shift invariance comes in when we argue for elementary equivalence - in fact, the relevant structures are isomorphic, via the "shift-by-$m$" map.)
For example, the unary relation naming the even numbers is $m$-shift invariant for every even $m$, while the function $a*b=\max\{a,b\}+\vert b-a\vert$ is $m$-shift invariant for every $m$ whatsoever.
I suspect this sort of minor tweak won't help you too much, but without knowing more about your particular goal it's not clear where to look for a better example. (If you're familiar with for