Recent experiments have demonstrated a method for extracting the depairing current of nanowires fabricated from thin-film dirty superconductors using the AC response of DC-current biased resonators. While the existing theoretical model for understanding this response, developed by Clem and Kogan, provides agreement with Eilenberger-Usadel theory at low temperatures, there is a systematic and substantial deviation from theory at elevated temperatures. We propose that the DC bias in the presence of electromagnetic oscillations leads to Joule heating in the superconductor. This heating, combined with the strong electron-electron scattering in these heavily disordered materials, leads to oscillations in the effective temperature of the superconductor which alter the kinetic inductance. In this work, we derive the expression for the shift in kinetic inductance in the presence of a bias current and demonstrate that this model provides a significantly improved agreement between experiment and theory. © 2023 American Physical Society.