A previously developed model of solute release from a swellable polymeric matrix induced by solvent uptake that obeys Fickian kinetics is here extended to cover the case of non-Fickian solvent uptake caused by slow structural relaxation of the swelling polymer. For this purpose, we have adopted a model description of the absorption process (also previously developed in our laboratory), which has been shown to be capable of realistically simulating a wide range of non-Fickian kinetics (including the well-known two-stage sorption and case II regimes). The aforementioned structural relaxation phenomena constitute important characteristics of glassy polymeric matrices. Accordingly, the predictions of the resulting combined model have been investigated in some detail in this series of articles. This particular article is concerned with the simplest version of the model, involving constant uptake kinetic parameters (diffusion coefficient and relaxation frequency) for both osmotically inactive and osmotically active solutes. Emphasis is put on possibilities of achieving and sustaining nearly constant release rates, and it is shown that such possibilities are considerably more numerous here than under conditions of Fickian solvent uptake kinetics.