TITLE: Singular models of K3 surfaces
QUESTION [5 upvotes]: Let us work over a ground field of characteristic zero. As is well-known, a K3 surface is a smooth projective geometrically integral surface $X$ whose canonical class $\omega_X$ is trivial and for which $\operatorname{H}^1(X,\mathscr{O}_X)$ vanishes.
A bit of folklore (proven e.g. in Beauville's Complex Algebraic Surfaces) is that if $X$ is a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^4$, or a smooth complete intersection of three quadrics in $\mathbb{P}^5$, then $X$ is K3.
The question
Suppose I have a (non-smooth) complete intersection $X_{2,3}$ of a quadric and a cubic in $\mathbb{P}^4$ all of whose singularities are rational double points, and a (non-smooth) complete intersection $X_{2,2,2}$ of three quadrics in $\mathbb{P}^5$, again with at most rational double points. 
Do these two surfaces have K3 surfaces as their minimal regular models?

I suspect the answer is yes; for example, since the singularities are assumed to be rational double points, they should admit a crepant resolution. However, I can't really locate this fact in the literature. As for the vanishing of the appropriate $\operatorname{H}^1$, I really haven't a clue.

REPLY [4 votes]: 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 It's in an unpublished preprint found here (p. 38 onwards).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.