In many of the existing refractive index (RI) sensing works, only the shape and size of plasmonic structures are usually taken into account, while the parameters of spacer layers are ignored. In this publication, we explored the long-range surface plasmon resonance (LRSPR) and Fabry-Pérot resonance coupling effects of our proposed gold nanoring cavity array/spacer layer/Au mirror/glass substrate. Both the RI sensitivity and full width at half-maximum (FWHM) values were superior than those of conventional surface plasmon resonance substrates. We discussed the tunability of the RI sensitivity through changing the RI and thickness of the spacer layer. Then, under the optimized parameter conditions of the spacer layer, the geometry parameters (including size, gap and periodicity) of gold nanoring cavity arrays were tuned to optimize the best RI sensitivity. Finally, we broke the structural symmetry of a nanoring cavity to introduce Fano resonances into our system, and a high RI sensitivity and figure-of-merit (FOM) of 695 nm per RIU (refractive index unit) and 96.5, respectively, were achieved when the breaking angle θ was 30°. This study opens up many possibilities for boosting the FOM of RI sensing by taking into account the hybridization effects of localized surface plasmon resonance, LRSPR, and Fabry-Pérot and Fano resonances.