Large birefringence and its electrical modulation by means of Fréedericksz transition makes nematic liquid crystals (LCs) a promising platform for tunable terahertz (THz) devices. The thickness of standard LC cells is in the order of the wavelength, requiring high driving voltages and allowing only a very slow modulation at THz frequencies. Here, we first present the concept of overcoupled metal-isolator-metal (MIM) cavities that allow for achieving simultaneously both very high phase difference between orthogonal electric field components and large reflectance. We then apply this concept to LC-infiltrated MIM-based metamaterials aiming at the design of electrically tunable THz polarization converters. The optimal operation in the overcoupled regime is provided by properly selecting the thickness of the LC cell. Instead of the LC natural birefringence, the polarization-dependent functionality stems from the optical anisotropy of ultrathin and deeply subwavelength MIM structures. The dynamic electro-optic control of the LC refractive index enables the spectral shift of the resonant mode and, consequently, the tuning of the phase difference between the two orthogonal field components. This tunability is further enhanced by the large confinement of the resonant electromagnetic fields within the MIM cavity. We show that for an appropriately chosen linearly polarized incident field, the polarization state of the reflected field at the target operation frequency can be continuously swept between the north and south pole of the Poincaré sphere. Using a rigorous Q-tensor model to simulate the LC electro-optic switching, we demonstrate that the enhanced light-matter interaction in the MIM resonant cavity allows the polarization converter to operate at driving voltages below 10 Volt and with millisecond switching times.