Recent X-ray studies revealed over-ionized recombining plasmas (RPs) in a dozen mixed-morphology (MM) supernova remnants (SNRs). However, the physical process of the over-ionization has not been fully understood yet. Here we report on spatially resolved spectroscopy of X-ray emission from W44, one of the over-ionized MM-SNRs, using XMM-Newton data from deep observations, aiming to clarify the physical origin of the over-ionization. We find that combination of low electron temperature and low recombination timescale is achieved in the region interacting with dense molecular clouds. Moreover, a clear anti-correlation between the electron temperature and the recombining timescale is obtained from each of the regions with and without the molecular clouds. The results are well explained if the plasma was over-ionized by rapid cooling through thermal conduction with the dense clouds hit by the blast wave of W44. Given that a few other over-ionized SNRs show evidence for adiabatic expansion as the major driver of the rapid cooling, our new result indicates that both processes can contribute to over-ionization in SNRs, with the dominant channel depending on the evolutionary stage.