Time-division multiplexing (TDM) multiple-input multiple-output (MIMO) radar systems have gained popularity in automotive applications, thanks to their high-resolution and cost-effective properties. However, when the relative motion between the target and the radar is significant and possibly fast time-varying, TDM MIMO systems suffer for reduced maximum unambiguous velocity and angle-Doppler coupling. In this work, a novel off-grid estimation method for TDM MIMO systems is developed that operates on a single cycle and is able to accurately estimate unambiguously the instantaneous velocity and the target direction of arrival (DOA). We model the angle-Doppler coupling through a space-time steering vector. Then, an iterative adaptive approach (IAA)-based method is exploited to reconstruct the 2D velocity-angle spectrum. Additionally, we account for off-grid errors when the targets do not lie on the search grids. A coarse-to-fine strategy is proposed to alleviate the computational burden, where a maximum likelihood (ML) estimator iteratively corrects the off-grid velocity and DOA estimates of each target. Simulated and experimental data are processed to demonstrate the effectiveness of the proposed method.