MIMO radars with electromagnetic vector sensor (EMVS) antennas (also known as EMVS-MIMO radars) have been investigated intensively for target direction estimation in recent years. Many existing works on this topic are based on an unrealistic assumption that each EMVS antenna transmits six identically polarized and mutually orthogonal waveforms. Furthermore, target-fluctuation induced non-coherent backscattering in polarization has been entirely overlooked. This paper aims to develop a general transmit-receive signal model for EMVS-MIMO radar direction estimation. Specifically, by taking transmit polarization agility and/or polarization non-coherent backscattering into account, the collected pulse signals are no longer of complete polarization, but of incomplete polarization. By regarding an incomplete polarized (IP) signal as two incoherent completely polarized (CP) signals with the same azimuth-elevation directions and drawing upon the idea of ESPRIT, a new azimuth-elevation direction estimation algorithm is thus derived. The advantage of the new algorithm is that it can provide closed-form, automatically paired azimuth-elevation direction estimates and require no information on the location/displacement of the transmit-receive antennas. Finally, simulations are conducted to demonstrate the effectiveness of the proposed algorithm.