Are there essentially undecidable theories the essential undecidability of which has not been established through a Gödel Sentence, that is, by diagonalization?
To the best of my knowledge, the answer is "no". All essentially undecidable theories seem to go back to Gödel's diagonalizazion technique. It is remarkable, or puzzling, that since the publication of Gödel's incompleteness article in 1931, the essential undecidability of some theories has not been established by other means than Gödel's diagonalization.
EDIT (not by OP): A theory is essentially undecidable if (it is consistent and) all its models have undecidable theories. This is equivalent to the (in my experience) more common term essential incompleteness, meaning that all recursively axiomatizable extensions of the theory are incomplete; see e.g. here.