I tried the commutator of the generators and it worked, but I had no real justification for making that computation. Is there a perspective from which trying the commutator is the "obvious" thing to think about?
I was inspired to try the commutator because of the following exercise: Let $x,y \in Sym(\Omega)$. If $\Gamma = supp(x) \cap supp(y)$, then $[x,y] \subseteq \Gamma \cup \Gamma^x \cup \Gamma^y$. In particular, if $|\Gamma|= 1$, then $[x,y]$ is a 3-cycle.
This is not exactly the same, however.
I'd be careful with statements about what is "obvious", but the way I came up with trying the commutator was: We want most of the elements to remain invariant and only a few of them to be permuted. Applying the generators $x,y$ and their inverses leaves the "bulk" of the elements unchanged if we have equal numbers of $x$ and $x^{-1}$ and equal numbers of $y$ and $y^{-1}$; then only elements at the "borders" will be perturbed. And the simplest non-trivial product of that sort is the commutator.