This is not so.
Let $R=\mathbb C$ and consider the module $F$ to be the finite sequences in $\ell^2({\mathbb C})$.
Let $F'$ be the submodule of all sequences $(z_n)$ with $\sum_nz_n=0$.
Then $F'$ is dense in $F$.
To see this, let $(w_n)$ be in $F$ and let $a=\sum_nw_n$.
Let $N$ be a natural number such that $w_n=0$ for $n\ge N$.
For $j\in\mathbb N$ let $z(j)$ be the element of $F'$ given by $z(j)_n=w_n$ if $n < N$ and $z(j)_n=-\frac 1j$ if $N\le j\le N+j-1$.
Then $\| z(j)-w\|=\sqrt{j |a|^2/j^2}=\sqrt{|a|^2/j}$ tends to zero, so $F'$ is dense in $F$.