In a nutshell, no, at least in the separable case.  Let $F\subseteq E^*$ be not norm dense, and with $F$ (norm) separable.  By Hahn-Banach there is $M\in E^{**}$ which is non-zero and annihilates $F$.  Let $f\in E^*$ with $\langle M,f \rangle=1$.

I shall use Helly's Lemma (which I have failed to find an online reference for; it follows from e.g. the [principle of local reflexivity](http://matwbn.icm.edu.pl/ksiazki/sm/sm100/sm10022.pdf)) which says that if $N\subseteq