Let $T$ be a complete countable theory with infinite models. The Baldwin-Lachlan theorem states that $T$ is uncountably categorical iff $T$ is $\omega$-stable and has no Vaughtian pair.
My question is somehow connected to the discussion on Morley's categoricity theorem which can be found here. I would be interested in seeing an example (either a simple exercise or a reference to a well-known result) of a situation where one can use this theorem to show that a certain theory is uncountably categorical. In other words, I'd like to see a $T$ which is not strongly minimal (because otherwise you don't need this theorem, and you can see directly the notion of dimension appear -- see Alex Kruckman's answer in the aforementioned post) and which can be proven to be uncountably categorical by proving its $\omega$-stability and its lack of Vaughtian pairs. For example, one could imagine a situation where, thanks to some quantifier elimination result, counting types over countable sets of parameters is not that difficult, and where for some trivial reason there cannot be a Vaughtian pair (for example, as in the case of real closed fields, because the cardinality of an infinite definable set is forced to be the same as that of the whole model).
In this post, two nontrivial examples of uncountably categorical theories are mentioned (the direct sum of infinitely many cyclic groups of order 4 and the example in the answer), but I was unable to prove that they are uncountably categorical, so I don't know if they would provide an answer to my question here.
Another related (less philosophical) question is the following: I've recently learned that $DCF_0$ (which is known to be $\omega$-stable) has uncountably many countable models. By a consequence of the Baldwin-Lachlan theorem, one can deduce that this theory is not uncountably categorical. Is it easy to exhibit a Vaughtian pair?