The dual semigroup is equivalent in norm to its original semigroup
Let $\psi_t$ be as in the question with $\|\phi\|\leq 1$. We have $\psi_t\in E^\ast$ and $$ \|\psi_t\|\leq \frac 1 t\int_0^t \|T_s'\phi\|\,ds\leq \frac 1 t\int_0^t\|T_s\|\,ds. $$ In particular, for every $\epsilon>0$ there exists $T\geq 0$ such that $\|\psi_t\|\leq M+\epsilon$ for $t<T$.
Thus $$ \phi(f)=\lim_{t\searrow 0}\psi_t(f)\leq (M+\epsilon)\sup\{\psi(f)\mid\psi\in E^\ast,\,\|\psi\|\leq 1\} $$ for every $\epsilon>0$, which implies $$ \|f\|=\sup\{\phi(f)\mid \phi\in E',\|\phi\|\leq1\}\leq M\sup\{\psi(f)\mid\psi\in E^\ast,\,\|\psi\|\leq 1\}. $$