We present a data-driven approach for calculating adjoint sensitivities in unsteady turbulent flows, with application to shape optimization and output-based adaptation. Such simulations are much more expensive than steady models, so that each function evaluation, usually a statistic such as a time-averaged output, requires orders of magnitude more computational time and resources than in steady-state. This cost impedes the application of unsteady simulations to many-query studies, such as optimization, particularly via gradient-free methods that rely on numerous function evaluations. Furthermore, the chaotic nature of unsteady turbulent simulations prevents the calculation of gradients via adjoint-based methods, which become unstable and require expensive regularization techniques to provide meaningful answers. Our approach does not use unsteady adjoint equations but instead relies on unsteady data to train a corrected turbulence model, which then yields the required adjoint solutions. It is non-intrusive and inexpensive, requiring only a small number of unsteady forward simulations, but sufficiently powerful to capture unsteady effects in the sensitivities. Results for high-order discretizations of the unsteady Navier-Stokes equations, augmented by a corrected Spalart-Allmaras turbulence closure with and without a transition model, demonstrate the ability of the approach to accurately compute sensitivities in problems with unsteady turbulence.