Counterexamples can even be found in a domain, by taking rings of higher dimension or singular rings—once you're no longer over a PID, ideals will suffice. Take $R = k[x,y]$, the module $M = R$ itself, and the submodule $M' = (x,y)$. Or, take the ring $S = k[x,y]/(x^3-y^2)$, the module $N = S$ itself, and the submodule $N' = (x,y)$.
Counterexamples can even be found in a domain, by taking rings of higher dimension. Take $R = k[x,y]$, the module $M = R$ itself, and the submodule $N = (x,y)$.