I conjecture that following statement is false, but I cannot construct a counterexample
Let $N$ be an $L$-structure, with $L$ countable. Let $p(x)\subseteq L$ be a type that is not isolated i.e., there is no $\varphi(x)\in L$ such that $N\models\forall x\ [\varphi(x)\rightarrow p(x)]$. Then there is an $M\preceq N$ such that $M\not\models\exists x\ p(x)$.
The statement is true if we require $N$ to be $\omega$-saturated (essentially, by the omitting types theorem).