  Let $S$ be the polynomial ring on the space of non-square generic matrices or
the space of odd-sized skew-symmetric matrices, and let $I$ be the
determinantal ideal of maximal minors or $\operatorname{Pf}$ the ideal of
sub-maximal Pfaffians, respectively. Using desingularizations and
representation theory of the general linear group we expand upon work of
Raicu--Weyman--Witt to determine the $S$-module structures of
$\operatorname{Ext}^j_S(S/I^t, S)$ and
$\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S)$, from which we get the
degrees of generators of these $\operatorname{Ext}$ modules. As a consequence,
via graded local duality we answer a question of Wenliang Zhang on the socle
degrees of local cohomology modules of the form $H^j_\mathfrak{m}(S/I^t)$.
