A charge carrier in the channel of an organic field-effect transistor (OFET) is coupled to the electric polarization of the gate in the form of a surface Fröhlich polaron [N. Kirova and M. N. Bussac, Phys. Rev. B 68, 235312 (2003)]. We study the effects of the dynamical field of polarization on both small-polaron hopping and trap-limited transport mechanisms. We present numerical calculations of polarization energies, band- narrowing effects due to polarization, hopping barriers, and interface trap depths in pentacene and rubrene transistors as functions of the dielectric constant of the gate insulator and demonstrate that a trap-and-release mechanism more appropriately describes transport in high-mobility OFETs. For mobilities on the order 0.1 cm^2/V s and below, all states are highly localized and hopping becomes the predominant mechanism.