We propose and demonstrate a new type of electro-optic polymeric microring resonators, where the shape of the transmission spectrum is controlled by losses and phase shifts induced at the asymmetric directional coupler between the cavity and the bus waveguide. The theoretical analysis of such Charon microresonators shows, depending on the coupler design, three different transmission characteristics: normal Lorentzian dips, asymmetric Fano resonances, and Lorentzian peaks. The combination of the active azo-stilbene based polyimide SANDM2 surrounded by the hybrid polymer Ormocomp allowed the first experimental demonstration of electro-optic modulation in Charon microresonators. The low-loss modulators (down to 0.6 dB per round trip), with a radius of 50 microm, were produced by micro-embossing and exhibit either highly asymmetric and steep Fano resonances with large 43-GHz modulation bandwidth or strong resonances with 11-dB extinction ratio. We show that Charon microresonators can lead to 1-V half wave voltage all-polymer micrometer-scale devices with larger tolerances to coupler fabrication limitations and wider modulation bandwidths than classical ring resonators.