We study the drift mobility of a molecular polaron in the presence of an external applied field and Coulomb traps. The model is based on one previously developed for geminate recombination of photogenerated charge carriers. It is shown that the unusual combination of Poole-Frenkel-like field dependence and non-Arrhenius temperature dependence of the mobility, measured experimentally in molecular films, is well reproduced by this model. Our key result is that this nearly universal experimental behavior of the mobility arises from competition between rates of polaron trapping and release from a very low density of Coulomb traps.