Marker-free and non-invasive detection of analyte traces in aqueous solution using integrated optical resonators is an emerging technique within bioanalytics. Here, we present a single-mode silicon-nitride stadium resonator operating at the red edge of the visible spectrum, showing sensitivities larger than 200 nm/RIU and transmission dips with extinction ratios up to 15 dB. We present a mathematical model that allows optimization of the resonator sensitivity using the mode properties only, revealing that optimal sensing conditions are achieved for TM-polarized modes close to modal cut-off. The short operation wavelength makes the system especially suitable for nanoparticle enhanced sensing potentially down to the single-molecule limit.