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I misread your question, so the comments above are answering a different question than you asked. I thought you were asking about the lengths of the sheaf and its dual sheaf, not the lengths of the supports. Here is the easiest counterexample I know for the lengths of the supports. Let $R$ be the local ring $k[x,y]_{\langle x,y \rangle}$, with maximal ideal $\mathfrak{m} = \langle x,y \rangle$. Let $E$ be the finite length module $R/\mathfrak{m}^2 \cong k[x,y]/\langle x^2,xy,y^2 \rangle$. Then the dual $\text{Ext}^2_R(E,\omega_R)$ is isomorphic to the cokernel of the $R$-module homomorphism, $$\phi:R^{\oplus 3} \to R^{\oplus 2}, \ \left[\begin{array}{r} a \\ b \end{array} \right] \mapsto \left[\begin{array}{rr} y & -x & 0 \\ 0 & y & -x \end{array} \right] \left[ \begin{array}{r} a \\ b \end{array} \right].$$ In particular, the support of $E$ is the ideal $\mathfrak{m}^2$ with cokernel $R/\mathfrak{m}^2$, yet the support of $E^D$ is the ideal $\mathfrak{m}$ with cokernel $R/\mathfrak{m}$.