The following two given examples give that the modules are Artinian but not Notherian.
- It is well-known that $\mathbb{Z}_{p^\infty}$ Prüfer $p$-group is an Artinian $\mathbb{Z}$-module but not Noetherian.
- Suppose that $R$ is a complete Noetherian local power series ring. Then, injective hull of residue field of $R$ (denoted by $E_{R}$) is an Artinian $R$-module but not Noetherian. Also, by Matlis duality, every submodule of $E_{R}$ will be annihilated an ideal of $R$, so it is a torsion module. Moreover, suppose that $R/I$ is a complete Noetherian local power series quotient ring, we have the same result.
Question: Could anyone provide an example of non-torsion Artinian module over unital commutative non-domain ring is not Noetherian module?
Unless I'm missing something, I think you can take $M$ to be any $R$ module that is Artinian but not Noetherian, and $N$ to be any Artinian, non-torsion (by your definition) $S$ module, then $M\times N$ as an $R\times S$ module has all the qualities you require.
You can take $R=\mathbb Z$ and $M=\mathbb Z_{p^\infty}$ and
$S=N=F_2[x,y]/(x,y)^2$
Since the regular elements of $R\times S$ are of the form $(r,s)$ with $r$ regular in $R$, $s$ regular in $S$, the element $x$ that doesn't have a regular annihlator in $S$ creates and element $(0,x)$ that doesn't have a regular annihilator in $R\times S$.
Of course, $R\times S$ is not a domain.