It is rather common for a Banach space to have a weaker norm for which the unit ball is complete. A historically important example is the $L^1$ norm on $L^\infty$.  This provides a counterexample to your conjecture---take for $E$ the space $L^\infty$ regarded as a subspace of its own dual in the usual way.  This example was used by Saks (more precisely, the fact that the unit ball of $L^\infty$ has the Baire property under the $L^1$ norm) in his proof of what is now known as the Viali-Hahn-Saks theorem.