In this paper, we propose a compact plasmonic sensor structure comprised of a metal-dielectric-metal (MDM) waveguide, and a baffle plate in waveguide core and two side-coupled rectangular cavities. In this structure, two Fano resonances are achieved and can be tuned independently by changing the structural parameters of the cavities. Especially, when the resonant wavelengths of the two Fano resonances are the same, the sensing sensitivity can be enhanced by coupling between two Fano resonances. By investigating the transmission spectrum, the effect of structural parameters on Fano resonances and the refractive index sensitivity of the sensor structure are analyzed in detail. The numerical simulations demonstrate a sensitivity as high as 1295nm/RIU and a figure of merit of 1647.