Organic mixed ionic-electronic conductors (OMIECs) directly convert between ionic and electronic charge through electrochemical (de)doping, enabling a wide range of applications in bioelectronics, neuromorphic computing, and energy storage and conversion. While both ionic and electronic transport are individually well characterized, their combined transport has been difficult to describe self-consistently. We use in situ measurements of electrochemical (de)doping of an archetypal OMIEC to inform a quasi-field drift-diffusion model, which accurately captures experimentally measured ion transport across a range of potentials. We find that the chemical potential of holes, which is modulated by changes in doping level, represents a major driving force for mixed charge transport. Using numerical simulations at device-relevant time scales and potentials, we find that the competition between hole drift and diffusion leads to diffuse space charge regions despite high charge densities. This effect is unique to mixed conducting systems where mobile ionic charges can compensate the accumulation or depletion of electronic charge, thereby screening electrostatic driving forces.