There has been significant recent interest in the arithmetic properties of the coefficients of $F(1-q)$ and $\mathscr{F}_t(1-q)$ where $F(q)$ is the Kontsevich-Zagier strange series and $\mathscr{F}_t(q)$ is the strange series associated to a family of torus knots as studied by Bijaoui, Boden, Myers, Osburn, Rushworth, Tronsgard and Zhou. In this paper, we prove prime power congruences for two families of generalized Fishburn numbers, namely, for the coefficients of $(ζ_N - q)^s F((ζ_N - q)^r)$ and $(ζ_N - q)^s \mathscr{F}_t((ζ_N - q)^r)$, where $ζ_N$ is an $N$th root of unity and $r$, $s$ are certain integers.