Unraveling the dominant charge transport mechanism in high-mobility amorphous oxide semiconductors is still a matter of controversy. In the present study we extended the random band-edge model suggested before for the charge transport and Hall-effect mobility in such disordered materials [Fishchuk et al., Phys. Rev. B 93, 195204 (2016)], and also describe the field-effect-modulated thermoelectricity in amorphous In-Ga-Zn-O ($a$-IGZO) films under the same premises. The model is based on the concept of charge transport through the extended states and assumes that the transport is limited by the spatial variation of the position of the band edge due to the disorder potential, rather than by localized states. The theoretical model is formulated using the effective medium approximation framework and describes well basic features of the Seebeck coefficient in disordered materials as a function of energy disorder, carrier concentration, and temperature. Carrier concentration dependencies of power factor and thermoelectric figure of merit have been also considered for such systems. Besides, our calculations reveal a remarkable turnover effect from a negative to a positive temperature dependence of Seebeck coefficient upon increasing carrier concentration. The suggested unified model provides a good quantitative description of available experimental data on the Seebeck coefficient and the charge mobilities measured in the same $a$-IGZO transistor as a function of the gate voltage and temperature by considering the same charge transport mechanisms. This promotes a deeper understanding and a more credible and accurate description of the transport process in $a$-IGZO films.