In this paper, we present the study of the acousto-optic behavior of underwater-acoustic sensors constituted by fiber Bragg gratings (FBGs) coated by ring-shaped overlays. Via full-wave numerical simulations, we study the complex opto-acousto-mechanical interaction among an incident acoustic wave traveling in water, the optical fiber surrounded by the ring shaped coating, and the FBG inscribed the fiber, focusing on the frequency range 0.5-30 kHz of interest for SONAR applications. Our results fully characterize the mechanical behavior of an acoustically driven coated FBG, and highlight the key role played by the coating in enhancing significantly its sensitivity by comparison with a standard uncoated configuration. Furthermore, the hydrophone sensitivity spectrum exhibits characteristic resonances, which strongly improve the sensitivity with respect to its background (i.e., away from resonances) level. Via a three-dimensional modal analysis, we verify that the composite cylindrical structure of the sensor acts as an acoustic resonator tuned at the frequencies of its longitudinal vibration modes. In order to evaluate the sensor performance, we also carry out a comprehensive parametric analysis by varying the geometrical and mechanical properties of the coating, whose results also provide a useful design tool for performance optimization and/or tailoring for specific SONAR applications. Finally, a preliminary validation of the proposed numerical analysis has been carried out through experimental data obtained using polymeric coated FBGs sensors revealing a good agreement and prediction capability.