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Kerr-microresonator frequency combs in integrated photonics waveguides are promising technologies for next-generation positioning, navigation, and timing applications, with advantages that include platforms that are mass-producible and CMOS-compatible and spectra that are phase-coherent and octave-spanning. Fundamental thermal noise in the resonator material typically limits the timing and frequency stability of a microcomb. The small optical mode volume of the microresonators exaggerates this effect, as it both increases the magnitude and shortens the timescale of thermodynamic fluctuations. In this work, we investigate thermal instability in silicon nitride microring resonators as well as techniques for reducing their effects on the microcomb light. We characterize the time-dependent thermal response in silicon nitride microring resonators through experimental measurements and finite element method simulations. Through fast control of the pump laser frequency, we reduce thermal recoil due to heating. Finally, we demonstrate the utility of a coupled microresonator system with tunable mode interactions to increase the stability of a soliton against thermal shifts.