Aims. Temperature uncertainties plague our understanding of abundance variations within the interstellar medium. Using the PHANGS-MUSE large program, we develop and apply a new technique to model the strong emission lines arising from H ii regions in 19 nearby spiral galaxies at ~50 pc resolution and infer electron temperatures for the nebulae. Methods. Due to the charge-exchange coupling of the ionization fraction of the atomic oxygen to that of hydrogen, the emissivity of the observed [O i]λ6300/Hα line ratio can be modeled as a function of the gas phase oxygen abundance (O/H), ionization fraction (f_ion), and electron temperature (T_e). We measure O/H using a strong-line metallicity calibration and identify a correlation between f_ion and [S iii]λ9069/[S ii]λ6716,6730, tracing ionization parameter variations. Results. We solve for T_e and test the method by reproducing direct measurements of T_e([N ii]λ5755) based on auroral line detections to within ~600 K. We apply this“charge-exchange method” of calculating T_e to 4129 H ii regions across 19 PHANGS-MUSE galaxies. We uncover radial temperature gradients, increased homogeneity on small scales, and azimuthal temperature variations in the disks that correspond to established abundance patterns. This new technique for measuring electron temperatures leverages the growing availability of optical integral field unit spectroscopic maps across galaxy samples, increasing the statistics available compared to direct auroral line detections.