From this question Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties? I learned that if $X$ is a smooth complex projective variety of dimension $g$, then $X$ is a torsor over an Abelian variety (its Albanese variety) if and only if $\omega_X \cong \mathcal{O}_X$ and ${\rm h^1}(X; \mathcal{O}_X) = g$.

I would like to know the corresponding statement over an algebraically closed field of positive characteristic. If necessary, we could exclude small primes:

By the Bombieri-Mumford classification of surfaces in positive characteristic, I know that the above statement holds for $g=2$ and characteristics different from 2 and 3. In those small characteristics, one also gets examples of "quasi-hyperelliptic surfaces" (essentially because the Albanese could be a non-reduced group scheme), and they distinguish the two classes via étale cohomology (which as far as I can tell, I cannot calculate for my examples of interest).

For my own purposes, I want the answer for $g=3$, but the answer in general is also welcome. If it helps, I also know that ${\rm h}^i(X; \mathcal{O}_X) = \binom{g}{i}$ for all $i$.