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Evidence of disequilibrium chemistry due to vertical mixing in the atmospheres of many T and Y-dwarfs has been inferred due to enhanced mixing ratios of CO and reduced NH$_3$. Atmospheric models of planets and brown dwarfs typically parameterize this vertical mixing phenomenon with the vertical eddy diffusion coefficient, $K_{\rm zz}$. While $K_{\rm zz}$ can perhaps be approximated in the convective regions in the atmosphere with mixing length theory, in radiative regions the strength of vertical mixing is uncertain by many orders of magnitude. With a new grid of self-consistent 1D model atmospheres from $T_{\rm eff}$ of 400 - 1000 K, computed with a new radiative-convective equilibrium python code PICASO 3.0, we aim to assess how molecular abundances and corresponding spectra can be used as a probe of depth-dependent $K_{\rm zz}$. At a given surface gravity, we find non-monotonic behavior in the CO abundance as a function of $T_{\rm eff}$, as chemical abundances are sometimes quenched in either of two potential atmospheric convective zones, or quenched in either of two possible radiative zones. The temperature structure and chemical quenching behavior also changes with gravity. We compare our models with available near-infrared and M-band spectroscopy of several T and Y-dwarfs and assess their atmospheric vertical mixing profiles. We also compare to color-magnitude diagrams and make predictions for JWST spectra. This work yields new constraints, and points the way to significant future gains, in determining $K_{\rm zz}$, a fundamental atmospheric parameter in substellar atmospheres, with significant implications for chemistry and cloud modeling.