We observe strong modal coupling between the TE_00 and TM_00 modes in Si_3N_4 ring resonators revealed by avoided crossings of the corresponding resonances. Such couplings result in significant shifts of the resonance frequencies over a wide range around the crossing points. This leads to an effective dispersion that is one order of magnitude larger than the intrinsic dispersion and creates broad windows of anomalous dispersion. We also observe the changes to frequency comb spectra generated in Si_3N_4 microresonators due to polarization mode and higher-order mode crossings and suggest approaches to avoid these effects. Alternatively, such polarization mode crossings can be used as a tool for dispersion engineering in microresonators.