A polynomial variable increases depth by one

Multiplication by $X$ on $M[X]$ is injective coefficientwise and its cokernel is $M$. These properties survive localization, so $X$ is $N$-regular and $N\mathrm{/}XN\cong M$.

As an $S$-module, $N\mathrm{/}XN$ is $M$ with $X$ acting by zero; regular sequences from $(\mathfrak{m},X)$ on it are exactly regular sequences from $\mathfrak{m}$ on $M$. Hence the regular-quotient depth formula gives ${\mathrm{depth}}_S(N)={\mathrm{depth}}_R(M)+1$.

Put $A=R\mathrm{/}{\mathrm{Ann}}_R(M)$. Then ${\mathrm{Supp}}_S(N)=\mathrm{Spec}(A[X]{)}_{({\mathfrak{m}}_A,X)}\mathrm{.}$ The Choice-qualified polynomial-dimension theorem applies under the given Axiom of Choice and gives $\dim A[X]=\dim A+1$; its lower chain is attained below $({\mathfrak{m}}_A,X)$ because $A$ is local. Therefore the displayed localization has dimension $\dim A+1={\dim }_R(M)+1$. This is the asserted dimension equality and does not require every support prime to be extended from $R$ or to contain $X$.