$\newcommand\si\sigma$Yes, at least for $p\ge1$. Note that 
$$F_p:=F_p(m,\si)=\|(\si-1)Z+mR||_p\le|\si-1|\,\|Z\|_p+|m|,$$
where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$. 

On the other hand, by Jensen's inequality, conditioning on $R$ and then on $Z$, we get 
$$F_p\ge\max(|\si-1|\,\|Z\|_p,|m|)\ge\frac12\,(|\si-1|\|Z\|_p+|m|).$$