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 Sr2CuTeO6 and Sr2CuWO6 are physical realizations of this model but have distinctly different types of magnetic order and interactions due to a d(10)/d(0) effect. Long-range magnetic order is suppressed in the solid solution Sr2CuTe1-xWxO6 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 present a comprehensive neutron scattering study of this system. We show using polarized neutron scattering that the spin liquid-like x = 0.2 and x = 0.5 samples have distinctly different local spin correlations, which suggests that they have different ground states. Low-temperature neutron diffraction measurements of the magnetically ordered W-rich samples reveal magnetic phase separation, which suggests that the previously ignored interlayer coupling between the square planes plays a role in the suppression of magnetic order at x approximate to 0.6. These results highlight the complex magnetism of Sr2CuTe1-xWxO6 and hint at a new quantum critical point between 0.2 < x < 0.4.