We use Ramsey spectroscopy to experimentally probe the quantum dynamics of disordered dipolar-interacting ultracold molecules in a partially filled optical lattice, and we compare the results to theory. We report the capability to control the dipolar interaction strength. We find excellent agreement between our measurements of the spin dynamics and theoretical calculations with no fitting parameters, including the dynamics dependence on molecule number and on the dipolar interaction strength. This agreement verifies the microscopic model expected to govern the dynamics of dipolar molecules, even in this strongly correlated beyond-mean-field regime, and represents the first step towards using this system to explore many-body dynamics in regimes that are inaccessible to current theoretical techniques.