Millimeter-wave (mmWave) joint communication-radar (JCR) will enable high data rate communication and high-resolution radar sensing for applications such as autonomous driving. Prior JCR systems that are based on the mmWave communications hardware, however, suffer from a limited angular field-of-view and low estimation accuracy for radars due to the employed directional communication beam. In this paper, we propose an adaptive and fast combined waveform-beamforming design for the mmWave automotive JCR with a phased-array architecture that permits a trade-off between communication and radar performances. To rapidly estimate the mmWave automotive radar channel in the Doppler-angle domain with a wide field-of-view, our JCR design employs circulant shifts of the transmit beamformer to acquire radar channel measurements and uses two-dimensional compressed sensing (CS) in the space-time dimension. We optimize these circulant shifts to minimize the coherence of the CS matrix, under the space-time sampling constraints in our problem. We evaluate the JCR performance trade-offs using a normalized mean square error (MSE) metric for radar estimation and a distortion MSE metric for data communication, which is analogous to the distortion metric in the rate-distortion theory. Additionally, we develop a MSE-based weighted average optimization problem for the adaptive JCR combined waveform-beamforming design. Numerical results demonstrate that our proposed JCR design enables the estimation of short- and medium-range radar channels in the Doppler-angle domain with a low normalized MSE, at the expense of a small degradation in the communication distortion MSE.