Why not just take a quintic threefold in projective four-space containing a projective plane, $L$. If the quintic is general, then there will be 16 ODPs along the plane. There are two small resolutions, one of which is obtained by blowing up the plane. This should work for sufficiently large $p$. It is easy to see these two models are not isomorphic.

This is a codimension one condition, so one has a nineteen dimensional family of such K3 surfaces. There will be a countable number of 18 dimensional subfamilies of algebraic elliptic K3 surfaces.

By the way, one thought occurs to me prompted by your mention of $E_8$. The CY with Hodge numbers 11 and 491 is easily described as a general elliptic fibration over the Hirzebruch surface $F_12$. If memory serves me right, this fibration has a curve of type $II^*$ fibres over the section with self-intersection $-12$. Of course this fibre has an extended $E_8$ Dynkin diagram.

Certainly very intriguing. I didn't realise that the curve was symmetric.

In the sactter plot, do the two parabola-like upper curves match up if one is translated to the other?

Lev, that curve has fascinated me for a long time. It was even visible in an earlier paper of Candelas et al on hypersurfaces in weighted projective space. To the best of my knowledge, noone has an explanation for it.