We recently developed a tensorial constitutive model for dense, shear-thickening particle suspensions that combines rate-independent microstructural evolution with a stress-dependent jamming threshold. This gives a good qualitative account for reversing flows, although it quantitatively overestimates structural anisotropy [J. J. J. Gillissen et al., Phys. Rev. Lett. 123(21), 214504 (2019)]. Here, we use the model to predict the unjamming effect of superposed transverse oscillations on a steady shear flow in the thickened regime [N. Y. C. Lin et al., Proc. Natl. Acad. Sci. U.S.A. 113, 10774 (2016)]. The model successfully reproduces the oscillation-mediated viscosity drop observed experimentally. We compare the time-dependent components of the stress and microstructure tensors to discrete-element simulations. Although the model correctly captures the main qualitative behavior, it generally over-predicts the microstructural anisotropy in steady shear, and it under-predicts the number of particle contacts in oscillating shear. It also does not fully capture the correct variation in phase angle between the transverse component of the microstructure and the shear rate oscillations as the amplitude of the latter is increased. These discrepancies suggest avenues for future improvements to the model.