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.)