  Optical frequency stabilization is a critical component for precision
scientific systems including quantum sensing, precision metrology, and atomic
timekeeping. Ultra-high quality factor photonic integrated optical resonators
are a prime candidate for reducing their size, weight and cost as well as
moving these systems on chip. However, integrated resonators suffer from
temperature-dependent resonance drift due to the large thermal response as well
as sensitivity to external environmental perturbations. Suppression of the
cavity resonance drift can be achieved using precision interrogation of the
cavity temperature through the dual-mode optical thermometry. This approach
enables measurement of the cavity temperature change by detecting the resonance
difference shift between two polarization or optical frequency modes. Yet this
approach has to date only been demonstrated in bulk-optic whispering gallery
mode and fiber resonators. In this paper, we implement dual-mode optical
thermometry using dual polarization modes in a silicon nitride waveguide
resonator for the first time, to the best of our knowledge. The temperature
responsivity and sensitivity of the dual-mode TE/TM resonance difference is
180.7$\pm$2.5 MHz/K and 82.56 $\mu$K, respectively, in a silicon nitride
resonator with a 179.9E6 intrinsic TM mode Q factor and a 26.6E6 intrinsic TE
mode Q factor. Frequency stabilization is demonstrated by locking a laser to
the TM mode cavity resonance and applying the dual-mode resonance difference to
a feedforward laser frequency drift correction circuit with a drift rate
improvement to 0.31 kHz/s over the uncompensated 10.03 kHz/s drift rate. Allan
deviation measurements with dual-mode feedforward-correction engaged shows that
a fractional frequency instability of 9.6E-11 over 77 s can be achieved.
