Singular models of K3 surfaces

Yes, this is true: the smooth minimal model is a $K3$ surface.

In fact, let $\bar{X}$ be the resolution of the singularities of $X$. Then the following holds.

(1) Rational double points impose no adjunction conditions to canonical forms, hence $\omega_{\bar{X}}$ is trivial.

(2) Rational double points have simultaneous resolution, hence $\bar{X}$ is deformation equivalent (and so diffeomorphic by Ehresmann's Theorem) to a smooth complete intersection $X^{sm}$ of the same type as $X$. This implies $b_1(\bar{X})=b_1(X^{sm})=0$, hence $H^1(\bar{X}, \, \mathcal{O}_{\bar{X}})=0$.

For part (1) you can look at M. Reid's Young Person Guide to Canonical Singularities, whereas part (2) can be found in Kollar-Mori's book Birational Geometry of Algebraic Varieties.

For what it's worth, I wrote up a proof (pretty detailed) following the hints in Francesco Polizzi's answer. It's in an unpublished preprint found here (p. 38 onwards). I am not a geometer, so the exposition may be a little clunky, but at least I think all the crucial steps are there.