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) which says that if $N\subseteq