We report a direct measurement of $D^0-overline{D}^0$ mixing parameters through a time-dependent amplitude analysis of the Dalitz plots of $D^0 \rightarrow K_S^0 \pi^+ \pi^-$ and, for the first time, $D^0 \rightarrow K_S^0 K^+ K^-$ decays. The low-momentum pion $\pi_s^+$ in the decay $D^{*+} \rightarrow D^0 \pi_s^+$ identifies the flavor of the neutral $D$ meson at its production. Using 468.5 fb$^{-1}$ of $e^+e^-$ colliding-beam data recorded near $\sqrt s = 10.6$~GeV by the BABAR detector at the PEP-II asymmetric-energy collider at SLAC, we measure the mixing parameters $x= [1.6 \pm 2.3 ({\rm stat.}) \pm 1.2 ({\rm syst.}) \pm 0.8 ({\rm model}) ] \times10^{-3}$, and $y= [5.7 \pm 2.0 ({\rm stat.}) \pm 1.3 ({\rm syst.}) \pm 0.7 ({\rm model}) ]\times 10^{-3}$. These results provide the best measurement to date of $x$ and $y$. The knowledge of the value of $x$, in particular, is crucial for understanding the origin of mixing. Show Partial Abstract