Here is a vague idea for how one might prove this in general. $M$ is flat if and only if, for all finitely generated ideals $\langle x_1, x_2, \ldots, x_n \rangle$ of $R$, we have $\mathrm{Tor}_1(M, R/\langle x_1, \ldots, x_n \rangle) = 0$. $N$ is torsion free if and only if, for all nonzero $x$ in $R$, we have $\mathrm{Tor}_1(N, R/x)=0$. Can we somehow get a relation between $\mathrm{Tor}_1(M, R/\langle x_1, \ldots, x_n \rangle)$ and $\mathrm{Tor}_1(M^{\otimes n}, R/(x_1 \cdots x_n))$?