I don't have an answer, but a couple of comments that may be useful:
The group $H^1({\cal O}_X)$ can be identified with the Zariski tangent space of ${\rm Pic}^0(X)$. Since the latter can be non-reduced in characteristic $p$ (in characteristic zero, this is impossible by a theorem of Cartier), the dimension of $H^1({\cal O}_X)$ may be larger than $\dim{\rm Alb}(X)=\dim {\rm Pic}^0(X)=b_1(X)/2$. So maybe you would want to ask whether $$ \omega_X\cong{\cal O}_X, b_1(X)=2g $$ implies that $X$ is a torsor over an Abelian variety. For example, quasi-hyperelliptic surfaces with non-reduced ${\rm Pic}^0(X)$ satisfy $\omega_X\cong{\cal O}_X$, $h^1({\cal O}_X)=2$, but their Albanese variety is $1$-dimensional.
So, let's assume that $\dim X=\dim {\rm Alb}(X)$, and let $f:X\to{\rm Alb}(X)$ be the Albanese morphism. By a result of Igusa, the pull-back map of global $1$-forms $$ f^*:H^0(\Omega^1_{\rm Alb X}) \rightarrow H^0(\Omega_X^1) $$ has no kernel. Now, if we knew that $f$ were generically finite, then it would be generically etale (since $\Omega_{\rm Alb(X)}^1$ is globally generated by $1$-forms and they survive the pull-back $f^*$ by Igusa's result just mentioned), and it also implies that $\Omega_X^1$ contains a trivial subbundle of rank $g$. The condition $\omega_X\cong{\cal O}_X$ then implies that $\Omega_X^1$ and thus, also the tangent bundle, are trivial bundles of rank $g$.
Varieties with trivial tangent bundles in characteristic $p$ were studied by Mehta, Nori, and Srinivas (Compositio Math. 64 (1987), no. 2, 191-212), and in case these varieties are moreover assumed to be ordinary (which one should think of as the "generic" or "nice" case), then there exists an Abelian variety $A$ and an etale Galois cover $A\longrightarrow X$.