$\newcommand\si\sigma$Yes. 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, for $p\ge1$, by Jensen's inequality, conditioning on $R$ and then on $Z$, we get $$F_p\ge\max(|\si-1|\,\|Z\|_p,|m|)\ge\frac12\,(|\si-1|\|Z\|_p+|m|).$$
Finally, for $p\in(0,1)$, by the central limit theorem and the Haagerup inequality, $$F_p\ge2^{1/2-1/p}(|\si-1|^2+|m|^2)^{1/2}\ge2^{-1/p}\,(|\si-1|+|m|).$$