I have what I am sure is a trivial question, but I can't seem to answer it for myself.
In model theory, there is a theorem of Hrushovski which shows that if T is a totally categorical theory (i.e., T is complete and has exactly one model of each infinite cardinality up to isomorphism), then (i) T is not finitely axiomatizable, but (ii) T is finitely axiomatizable modulo infinity; that is, there is some sentence p such that T is precisely the set of sentences true in every infinite model of p.
My question is to what extent the converse holds. That is, let T be a (EDIT: complete) theory which is finitely axiomatizable modulo infinity, but which is not finitely axiomatizable. Then is T necessarily totally categorical, and if not, what sort of assumptions on T are enough to ensure total categoricity?
(The assumption that T is not actually finitely axiomatizable is clearly necessary: otherwise, take the theory DLO of dense linear orders without endpoints, which is countable categorical but not uncountable categorical.)
Assuming that your logic includes equality, the answer to the first part of the question is "no".
For a counterexample, let $T$ be the theory with axioms $$(\exists x_1)(\exists x_2)\cdots(\exists x_n) \bigwedge_{1\le i<j\le n} x_i\ne x_j$$ for all integers $n\ge 2$. Then the models of $T$ are exactly all infinite interpretations of its language. This is not finitely axiomatizable, because in the pure predicate calculus with equality, any sentence with an infinite model also has finite models.
However, $T$ is finitely axiomatizable modulo infinity (you can take $p$ to be any propositional tautology, for example), and -- provided we add some relation apart from equality to its vocabulary -- it is obviously very far from from categorical; so far that it seems doubtful that there is any natural way to extend your condition to a sufficient one, without the additional conditions being sufficient in themselves.