Sufficiently disordered metals display systematic deviations from the behavior predicted by semi-classical Boltzmann transport theory. Here the scattering events from impurities or thermal excitations can no longer be considered as additive-independent processes, as asserted by Matthiessen's rule following from this picture. In the intermediate region between the regime of good conduction and that of insulation, one typically finds a change of sign of the temperature coefficient of resistivity, even at elevated temperature spanning ambient conditions, a phenomenology that was first identified by Mooij in 1973. Traditional weak coupling approaches to identify relevant corrections to the Boltzmann picture focused on long-distance interference effects such as "weak localization", which are especially important in low dimensions (1D and 2D) and close to the zero-temperature limit. Here we formulate a strong-coupling approach to tackle the interplay of strong disorder and lattice deformations (phonons) in bulk three-dimensional metals at high temperatures. We identify a polaronic mechanism of strong disorder renormalization, which describes how a lattice locally responds to the relevant impurity potential. This mechanism, which quantitatively captures the Mooij regime, is physically distinct and unrelated to Anderson localization, but realizes early seminal ideas of Anderson himself, concerning the interplay of disorder and lattice deformations.