The nature of dark matter remains unknown, but upcoming measurements probing the high-redshift Universe may provide invaluable insight. In the presence of dark matterbaryon scattering, the suppression in the matter power spectrum and the colder mean gas temperature are expected to modify the evolution of cosmic dawn and reionization. However, the contributions from such interactions to the baryon and dark matter temperature perturbations have been neglected thus far. In this work, we derive these contributions, evolve the cosmological perturbations until the end of the dark ages and show that they may have a significant impact in the beginning of cosmic dawn. In particular, we find that the amplitude of the temperature power spectrum at large scales can change by up to 1-2 orders of magnitude and that the matter power spectrum is further suppressed with respect to ΛCDM by 5-10% at k ∼ 200 Mpc −1 compared to the computation ignoring these contributions for scattering cross sections at current CMB limits. As a case example, we also compute the HI power spectrum from the dark ages, finding significant differences due to the changes in the temperature and ionization fraction power spectra. We argue that these new contributions must be included in studies of this dark matter model relying on cosmic dawn and reionization observables.