$\newcommand\spec{\operatorname{spec}}\newcommand\P{\mathbb P}\newcommand\msO{\mathscr O}\newcommand\push[1]{{#1}_*}$ Say I have some relative Weierstrass curve $f:W\to S$ (so $f$ is flat and proper of finite presentation, it's geometric fibers are integral of genus 1, and there's a section $\sigma:S\to W$ landing in the locus where $f$ is smooth). Feel free to assume $S=\spec k$ if it simplifies things below.
Locally on the base, $W$ is of the form $$\left\{Y^2Z+a_1XYZ+a_3YZ^2 = x^3+a_2X^2Z+a_4XZ^2+a_6Z^3\right\}\subset\P^2_U\text{ with }a_i\in\Gamma(U,\msO_U)$$ (where $U\subset S$ some small open), and given this description one can write down the expression $$\alpha:=\frac{dx}{2y+a_1x+a_3}.$$ My question is about the right way to interpret this expression; what sort of object is it?
If $f$ is smooth, then I want to think of $\alpha$ as a section (over $U$) of the Hodge bundle $\push f\Omega^1_{W/S}$; it's really giving me some regular differential form on my curve over $U$. When $f$ is not smooth, I am under the impression that the sheaf of 1-forms is no longer the right object to have in mind, and I would like to be able to interpret $\alpha$ as a section (again, over $U$) of $\push f\omega_{X/S}$, where $\omega_{X/S}$ is the relative dualizing sheaf of $W\xrightarrow fS$. However, I do not know a concrete enough description of dualizing sheaves in order to make this rigorous.
It is maybe worth mentioning that (unless I'm mistaken) $\alpha$ still defines a section in $\Gamma(U^{sm},\push f\omega_{X/S})$ above the locus in $U$ where $f$ is smooth, but I really want a section over all of $U$ (and am not sure how to extend this over the singular part).
Is there a way of using $\alpha$ to define a section of $\push f\omega_{X/S}$? If not, is this expression just less useful in the presence of singularities, or is it just the case that the right way to think of it is not as a section of the Hodge bundle?