We measure and, for the first time, theoretically predict four prototypical aqueous-drug diffusion coefficients in five soft-contact-lens material hydrogels where solute-specific adsorption is pronounced. Two-photon fluorescence confocal microscopy and UV/Vis-absorption spectrophotometry assess transient solute concentration profiles and concentration histories, respectively. Diffusion coefficients are obtained for acetazolamide, riboflavin, sodium fluorescein, and theophylline in 2-hydroxyethyl methacrylate/methacrylic acid (HEMA/MAA) copolymer hydrogels as functions of composition, equilibrium water content (30-90%), and aqueous pH (2 and 7.4). At pH2, MAA chains are nonionic, whereas at pH7.4, MAA chains are anionic (pKa≈5.2). All studied prototypical drugs specifically interact with HEMA and nonionic MAA (at pH2) moieties. Conversely, none of the prototypical drugs adsorb specifically to anionic MAA (at pH7.4) chains. As expected, diffusivities of adsorbing solutes are significantly diminished by specific interactions with hydrogel strands. Despite similar solute size, relative diffusion coefficients in the hydrogels span several orders of magnitude because of varying degrees of solute interactions with hydrogel-polymer chains. To provide a theoretical framework for the new diffusion data, we apply an effective-medium model extended for solute-specific interactions with hydrogel copolymer strands. Sorptive-diffusion kinetics is successfully described by local equilibrium and Henry's law. All necessary parameters are determined independently. Predicted diffusivities are in good agreement with experiment.