Edit: When writing this answer, I apparently forgot about the potential $p$, so the current version of the answer is only valid for $p=0$. I'll try to fix this as soon as I have time.
Yes.
If $q$ is globally Lipschitz on $\mathbb{R}^n$ with Lipschitz constant $L$, then all solutions of the ODE $\dot x = q(x)$ on $\mathbb{R}^n$ exist globally and, if $(\varphi_t)_{t \in \mathbb{R}}$ denotes the induced flow in $\mathbb{R}^n$, each function $\varphi_t: \mathbb{R}^n \to \mathbb{R}^n$ is Lipschitz continuous with constant $e^{|t|L}$ (this follows from Grönwall's lemma).
The solution of your PDE is given by $$ f(t,x) = f_0(\varphi_t(x)) $$ for each $t$ and $x$. So if $f_0$ decays faster than every polynomial, than so does $f(t, \cdot)$ for each fixed time $t$.