I believe this is false. There are equisingular families that are not étale locally, nor even formally locally, trivial. Let $(z,w)$ be coordinates in $\mathbb{A}^2$. Let $t$ be a coordinate on some dense Zariski open subset $V$ of $\mathbb{A}^1$. Consider the zero scheme $X$ of $zw(w-z)(w-tz)$ in $\mathbb{A}^2\times V$. I believe that $\Omega_X$ will have a direct sum decomposition as above for $V$ a suitable open subset. However, I very much doubt there is any étale decomposition of $X$. If there were, then the fiber of the projection to $Y$ would correspond to the zero locus of $(z,w)$. However, the formal completion of $X$ along this zero scheme is not a product, not even after making a further étale base change of $V$.
Edit. I just did the computation for my polynomial above. There is not a direct sum decomposition of $\Omega_X$. I was wrongly assuming that the existence of a direct sum decomposition would be preserved under small deformations. Thus, deforming from a trivial family to an equisingular family would give a counterexample. However, because the Ext group is nonzero, existence of a direct sum decomposition is not preserved under small deformations.