One example is given by Enriques surfaces in characteristic $2$. There are three types depending on the value of $\mathrm{Pic}^\tau$ (as a group scheme) which can be either $\mathbb Z/2$, $\mu_2$ or $\alpha_2$. In the first case $\omega_X$ is the generator so in particular it is non-trivial and $H^2(X,\mathcal O_X)=0$ (using of course Serre duality). In the two other cases $\omega_X$ is trivial (as it is numerically trivial and all numerically trivial line bundles are trivial) so that $h^2(X,\mathcal O_X)=1$. Now, $\alpha_2$ can be deformed to both $\mathbb Z/2$ and $\mu_2$ and such deformations can be lifted to deformations of Enriques surfaces (in fact $\mathrm{Pic}^\tau$ is flat in families of Enriques surfaces and the functor from deformations of the surfaces to those of $\mathrm{Pic}^\tau$ is formally smooth - Liedtke: arXiv:1007.0787, Ekedahl-Shepherd-Barron: unpublished). If we pick a connected family of Enriques surfaces with some special value being an $\alpha_2$-surface and generically $\mathbb Z/2$.

Such an example can be constructed (very) explicitly without deformation theory. Here is a semi-explicit construction which works in any positive characteristic. Fixa a a group scheme $A$ of order $p$ over $\mathbb A^1$ localised at $0$ (say) which is $\alpha_p$ at $0$ and $\mathbb Z/p$ elsewhere. By the Godeaux construction (which Raynaud showed works for such families) there is a free action of $A$ on a flat complete intersection $Y$ (of any dimension, which we assume is $\ge 2$) such that $X=Y/A$ is smooth (note that contrary to the case of an étale group scheme $Y$ will not be smooth). As $Y$ is a complete intersection we have that $\mathrm{Pic}^\tau_Y=0$ and from that it follows that $\mathrm{Pic}^\tau_X=A$. Now, $H^1(-,\mathcal O_-)$ is the tangent space of $\mathrm{Pic}^\tau$ so it is zero outside of $0$ and $1$-dimensional at $0$. (By being careful one can get Enriques surfaces for $p=2$, this I guess was the inspiration for the Godeaux construction).

One example is given by Enriques surfaces in characteristic $2$. There are three types depending on the value of $\mathrm{Pic}^\tau$ (as a group scheme) which can be either $\mathbb Z/2$, $\mu_2$ or $\alpha_2$. In the first case $\omega_X$ is the generator so in particular it is non-trivial and $H^2(X,\mathcal O_X)=0$ (using of course Serre duality). In the two other cases $\omega_X$ is trivial (as it is numerically trivial and all numerically trivial line bundles are trivial) so that $h^2(X,\mathcal O_X)=1$. Now, $\alpha_2$ can be deformed to both $\mathbb Z/2$ and $\mu_2$ and such deformations can be lifted to deformations of Enriques surfaces (in fact $\mathrm{Pic}^\tau$ is flat in families of Enriques surfaces and the functor from deformations of the surfaces to those of $\mathrm{Pic}^\tau$ is formally smooth - Liedtke: arXiv:1007.0787, Ekedahl-Shepherd-Barron: unpublished). If we pick a connected family of Enriques surfaces with some special value being an $\alpha_2$-surface and generically $\mathbb Z/2$, then we get an example.

Such an example can be constructed (very) explicitly without deformation theory. Here is a semi-explicit construction which works in any positive characteristic. Fixa a a group scheme $A$ of order $p$ over $\mathbb A^1$ localised at $0$ (say) which is $\alpha_p$ at $0$ and $\mathbb Z/p$ elsewhere. By the Godeaux construction (which Raynaud - Prop. 4.2.3, p-torsion du schema de Picard, Astérisque 64 - showed works for such families) there is a free action of $A$ on a flat complete intersection $Y$ (of any dimension, which we assume is $\ge 2$) such that $X=Y/A$ is smooth (note that contrary to the case of an étale group scheme $Y$ will not be smooth). As $Y$ is a complete intersection we have that $\mathrm{Pic}^\tau_Y=0$ and from that it follows that $\mathrm{Pic}^\tau_X=A$. Now, $H^1(-,\mathcal O_-)$ is the tangent space of $\mathrm{Pic}^\tau$ so it is zero outside of $0$ and $1$-dimensional at $0$. (By being careful one can get Enriques surfaces for $p=2$, this I guess was the inspiration for the Godeaux construction).