Supposing $R$ has zero divisors, there are elements $a,b \neq 0_R$ such that $ab = 0_R$.
Letting $m\in M$ be arbitrary, it seems to be the case that $$ ab \cdot m = 0_R \cdot M = 0_M. $$
So as long as there is at least one zero divisor in $R$, every element of $M$ is torsion. Is this in fact true, or is there a subtlety that I am missing?