... of course, these effects were known to Igusa and Serre already, but these examples are maybe more explicit

For example, William Lang ('Classical Godeaux Surface in Characteristic p', Math. Ann. 256, 419-427 (1981)) and Rick Miranda ('Nonclassical Godeaux Surfaces in Characteristic Five', Proc. AMS 91, 9-11 (1984)) address this. I'm not sure whether their examples are defined over the integers as they choose generic hyperplane sections, but maybe a computer search can establish this by hand? Miranda's examples also show explicitly that the Picard scheme need not be smooth - it becomes non-reduced for p=5 (in fact ${\rm Pic}^0$ is isomorphic to the non-reduced scheme $\mu_5$) ...

you might want to have a look at Jouanolou's book 'Theoremes de Bertini et applications'

Did you have a look at both of Shepherd-Barron's papers in Invent. Math. 106 (1991)? They're about surfaces, but go in the direction you're looking for...

nice! I was hoping for such a more elegant/elementary argument!

I've never thought about this, but you might want to look for a $\mathbb{Z}/4\mathbb{Z}$-action $\psi$ on $\tilde{X}$ such that $\psi\tau=\tau\psi$ (then, it will descend to $X$), and such that $\psi$ acts on global sections of $\omega_{\tilde{X}}$ via multiplication by $\sqrt{-1}$.

of course, $\mathbb{Z}/4\mathbb{Z}$ may act on a K3 surface. BUT: this action is supposed to be free (!), and this is impossible: the hypothetical quotient $S$ would have $\chi({\mathcal O}_S)= \chi({\mathcal O}_{\tilde{X}})/4=\frac{1}{2}$... I don't understand the other objection: an involution on $\tilde{X}$ acts on global sections of $\omega_X$ by ${\rm id}$ or by $-{\rm id}$. Since the map $H^0(\omega_{\tilde{X}})^{\otimes 2}\rightarrow H^0(\omega_{\tilde{X}}^{\otimes 2})$ is onto, we conlude that every involution acts trivially on global sections of $\omega_{\tilde{X}}^{\otimes2}$!

we conclude that $\sigma$ acts trivially on global sections of $\omega_X^{\otimes2}$... Is there no easier way to see this?!

Now, if we had $\tilde{\sigma}^2=\tau$, then this would give rise to a free $\mathbb{Z}/4\mathbb{Z}$-action on the K3 surface $\tilde{X}$, which is absurd. Thus, $\tilde{\sigma}$ is an involution on $\tilde{X}$. Since every section of $\omega_{\tilde{X}}^{\otimes2}$ arises as square of a section of $\omega_{\tilde{X}}$, this implies that $\tilde{\sigma}$ acts trivially on global sections of $\omega_{\tilde{X}}^{\otimes2}$. Now, every global section of $\omega_X^{\otimes2}$ pulls back to a global section of $\omega_{\tilde{X}}^{\otimes2}$ and by compatibility of the actions,

Hmm. I really thought there should be an easy argument, but I'm convinced that my arguments are too simple minded. Before editing further, what about the following? Let $\tilde{X}\to X$ be the associated K3 cover, and denote by $\tau$ the associated involution on $\tilde{X}$. Then, ${\rm Aut}(X)$ is isomorphic to ${\rm Aut}(\tilde{X},\tau)$, where this latter group is $\{ \varphi\in{\rm Aut}(X), \varphi \tau=\tau\varphi \}$ modulo $\tau$. In particular, the involution $\sigma$ lifts to an automorphism $\tilde{\sigma}$ of $\tilde{X}$.

However, the converse is false in positive characteristic: Igusa has given an example of a variety with trivial tangent bundle in characteristic $2$ that is not an Abelian variety: it arises as $(E\times E')/({\mathbb{Z}/2\mathbb{Z})$, where $E$ and $E'$ are elliptic curves, and the group $\mathbb{Z}/2\mathbb{Z}$ acts via sign involution on one factor and via translation of a $2$-torsion point on the other factor.