Plasmonic chemical and biological sensors offer significant advantages such as really compact sizes and extremely high sensitivity. Biosensors based on plasmonic waveguides and resonators are some of the most attractive candidates for mobile and wearable devices. However, high losses in the metal and complicated schemes for practical implementation make it challenging to find the optimal configuration of a compact plasmon biosensor. Here, we propose a novel plasmonic refractive index sensor based on a metal strip waveguide placed under a waveguide-based racetrack ring resonator made of the same metal. This scheme guarantees effective coupling between the waveguide and resonator and low loss light transmittance through the long-range waveguide. The proposed device can be easily fabricated (e.g., using optical lithography) and integrated with materials like graphene oxide for providing adsorption of the biomolecules on the sensitive part of the optical elements. To analyze the properties of the designed sensing system, we performed numerical simulations along with some analytical estimations. There is one other interesting general feature of this sensing scheme that is worth pointing out before looking at its details. The sensitivity of the considered device can be significantly increased by surrounding the resonator with media of slightly different refractive indices, which allows sensitivity to reach a value of more than 1 μm per refractive index unit.