Suppose that $T$ is the first-order theory of a given class of models $C$ over some signature.
Suppose $M$ is an arbitrary model of $T$. Does it follow that there is always a model $M'\in C$ and homomorphism $h:M' \to M$?
I suspect this is false but I would like to have a counterexample.
I am assuming that when you say $C$ is complete for $T$, you mean that $\phi \in T$ iff $M \models \phi$ for every $M \in C$. If so then your suspicion is true and the conjecture is false: take $T$ to be theory of real-closed fields, take $C = \{\Bbb{R}\}$ and take $\cal A$ to be field of algebraic real numbers. Then $C = \{\Bbb{R}\}$ and $\cal A$ satisfy precisely the sentences in $T$, but there can be no homomorphism from $\Bbb{R}$ to $\cal A$