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We describe searches for $B$ meson decays to the charmless vector-vector final states $\ensuremath{\omega}{K}^{*}$ and $\ensuremath{\omega}\ensuremath{\rho}$ in $89\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs produced in ${e}^{+}{e}^{\ensuremath{-}}$ annihilation at $\sqrt{s}=10.58\text{ }\text{ }\mathrm{G}\mathrm{e}\mathrm{V}$. We measure the following branching fractions in units of ${10}^{\ensuremath{-}6}$: $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}{K}^{*0})={3.4}_{\ensuremath{-}1.6}^{+1.8}\ifmmode\pm\else\textpm\fi{}0.4(<6.0)$, $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{K}^{*+})={3.5}_{\ensuremath{-}2.0}^{+2.5}\ifmmode\pm\else\textpm\fi{}0.7(<7.4)$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\rho}}^{0})={0.6}_{\ensuremath{-}1.1}^{+1.3}\ifmmode\pm\else\textpm\fi{}0.4(<3.3)$, and $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\rho}}^{+})={12.6}_{\ensuremath{-}3.3}^{+3.7}\ifmmode\pm\else\textpm\fi{}1.6$. The first error quoted is statistical, the …