You have the right idea to prove that the theory is totally transcendental by proving that it is $\omega$-stable. But this is a non-trivial component of Morley's theorem, and one that you're not going to be able to prove just by playing around with Löwenheim-Skolem.
It's also a step that will be included in any proof of Morley's theorem (e.g. it's Corollary 5.2.10 in Marker, Theorem 5.2.4 in Tent and Ziegler, and Theorem 3.8 in Morley's original paper). Maybe "I am studying Morley's theorem" means that you're trying to reprove the whole thing on your own! But if not, you're probably reading from some reference that includes this proof.
In any case, the idea is this: You show that for every every countable theory $T$ with infinite models and every infinite cardinal $\kappa$, there is a model $M\models T$ which only realizes countably many types over any countable set of parameters. How? You build an Ehrenfeucht-Mostowski model generated by a sequence of indiscernibles ordered like the ordinal $\kappa$ (this is the tricky part of the proof, which would take far too long to explain in this answer). But on the other hand, if there is a countable model $N$ of $T$ such that $|S_1(N)|\geq \aleph_1$, then you can realize a set of $\aleph_1$-many types over $N$ in an elementary extension $N\preceq N'$ of size $\kappa$. These two models $M$ and $N'$ aren't isomorphic, so $T$ isn't $\kappa$-categorical.