This is an immediate application of the behavior of dualizing complexes relative to finite morphisms (such as closed immersions).

To explain this, first
recall that if $f:Y \rightarrow Z$ is a finite morphism between noetherian schemes and $\omega$ is a dualizing complex on $Z$ then $f^{!}(\omega)$ is a dualizing complex on $Y$, where the functor $f^{!} = \mathbf{R}\mathscr{H}om_Z(f_{\ast}(O_Y),\cdot)$ on $D^+_{\rm{qc}}(Z) = D({\rm{QCoh}}(Z))$ is viewed with values in $D^+_{\rm{qc}}(Y) = D^+({\rm{QCoh}}(Y))$ in the evident manner.
Recall also that dualizing complexes are of Zariski-local nature, to be used tacitly below.

Now consider a Cohen-Macaulay scheme $Z$ with pure dimension $n\ge 1$, so a "normalized" dualizing complex on $Z$ (i.e., a dualizing complex whose associated "codimension" function coincides with the usual one) is given by $\omega_Z[n]$ for dualizing sheaves $\omega_Z$ on $Z$ (which in turn are unique up to tensoring against a line bundle). Suppose $a \in O_Z(Z)$ is $O_Z$-regular (i.e., nowhere a zero-divisor on $O_Z$) and vanishes somewhere, so $Y := V(a) \subset Z$ is non-empty and CM of pure dimension $n-1$. Then for the inclusion $j:Y \hookrightarrow Z$, a "normalized" dualizing complex $\omega_Y[n-1]$ on $Y$ is given by $j^{!}(\omega_Z[n])$. This says that $\mathscr{E}xt^i_Z(j_{\ast}O_Y, \omega_Z)$ vanishes for $i \ne 1$ and that the quasi-coherent $j_{\ast}O_Y$-module $\mathscr{E}xt^1_Z(j_{\ast}O_Y, \omega_Z)$ is a dualizing sheaf on $Y$ when "viewed" as a quasi-coherent $O_Y$-module.

The long exact sequence for $\mathscr{E}xt^{\bullet}_Z(\cdot, \omega_Z)$'s arising from the short exact sequence
$$0 \rightarrow O_Z \stackrel{a}{\rightarrow} O_Z \rightarrow j_{\ast}O_Y \rightarrow 0$$ now yields that $\omega_Z$ has vanishing $a$-torsion (i.e., $a$ is $\omega_Z$-regular) and that $\omega_Z/(a \cdot\omega_Z) \simeq \mathscr{E}xt^1_Z(j_{\ast}O_Y, \omega_Z) =: \omega_Y$ is a dualizing sheaf on $Y$. This sets up an inductive formalism that we shall now use.

In the original setup with a CM scheme $X$ of finite type over a field $k$ (which may be arbitrary) such that $X$ has pure dimension $n$, we can run such a slicing argument $n$ times on a sufficiently small Zariski-open neighborhood of any given closed point $x$ of $X$ (upon "spreading out" a system of parameters in the $n$-dimensional CM local ring $O_{X,x}$) to deduce that any system of parameters in the maximal ideal of $O_{X,x}$ is a regular sequence for the stalk $\omega_{X,x}$ of any dualizing sheaf $\omega_X$ for $X$ (such as a relative dualizing sheaf $\omega_{X/k}$, since $k$ is regular). Thus, $\omega_X$ has CM stalks at all closed points, so it is CM at all points (by localization) and hence is a CM coherent sheaf.

[An alternative approach would be to use the link to local duality on stalks to carry out arguments directly on Spec($O_{X,x}$) without needing to perform spreading-out, but the above seems more direct given the context of the question as posed.]