Understanding and minimizing the sources of frequency noise in nanomechanical resonators is crucial for many sensing applications. In this work, we report an ultracoherent perimeter-mode nanomechanical resonator co-integrated with an on-chip optical cavity. This device combines low thermomechanical force noise and low detector noise, allowing us to study its intrinsic frequency fluctuations in detail. We find that the fluctuations of two mechanical modes are strongly correlated. Moreover, we demonstrate the generation of a signal at the frequency difference between the two modes directly on chip via nonlinear optomechanical transduction. This `difference signal' has vastly reduced intrinsic frequency fluctuations and can be used for frequency tracking with high precision, as we establish in a proof-of-principle experiment.