  Quantum spin liquids are highly entangled magnetic states with exotic
properties. The $S = 1/2$ square-lattice Heisenberg model is one of the
foundational models in frustrated magnetism with a predicted, but never
observed, quantum spin liquid state. Isostructural double perovskites
Sr$_2$CuTeO$_6$ and Sr$_2$CuWO$_6$ are physical realizations of this model, but
have distinctly different types magnetic order and interactions due to a
$d^{10}/d^0$ effect. Long-range magnetic order is suppressed in the solid
solution Sr$_2$CuTe$_{1-x}$W$_x$O$_6$ in a wide region of $x = 0.05-0.6$, where
the ground state has been proposed to be a disorder-induced spin liquid. Here
we show that the spin-liquid-like $x = 0.2$ and $x = 0.5$ samples have
distinctly different local spin correlations, which suggests they have
different ground states. Furthermore, the previously ignored interlayer
coupling between the square-planes is likely to play a role in the suppression
of magnetic order on the W-rich side at $x \approx 0.6$. These results
highlight the complex magnetism of Sr$_2$CuTe$_{1-x}$W$_x$O$_6$ and hint at a
new quantum critical point at $x \approx 0.3$.
