We report experimental studies of a driven spin-mechanical system, in which a nitrogen-vacancy ($\mathrm{N}\text{\ensuremath{-}}$V) center couples to out-of-plane vibrations of a diamond cantilever through the excited-state deformation potential. Photoluminescence excitation studies show that in the unresolved sideband regime and under strong resonant mechanical driving, the excitation spectra of a $\mathrm{N}\text{\ensuremath{-}}$V optical transition feature two spectrally sharp peaks, corresponding to the two turning points of the oscillating cantilever. In the limit that the strain-induced frequency separation between the two peaks far exceeds the $\mathrm{N}\text{\ensuremath{-}}$V zero-phonon linewidth, the spectral position of the individual peak becomes sensitive to minute detuning between the mechanical resonance and the external driving force. For a fixed optical excitation frequency near the $\mathrm{N}\text{\ensuremath{-}}$V transition, $\mathrm{N}\text{\ensuremath{-}}$V fluorescence as a function of mechanical detuning features resonances with a linewidth that can be orders of magnitude smaller than the intrinsic linewidth of the mechanical mode. This enhanced sensitivity to mechanical detuning can potentially provide an effective mechanism for mechanical sensing, for example, mass sensing via measurements of induced changes in the mechanical oscillator frequency.