Drying of polymeric coatings often occurs under conditions where the relaxation time of polymer molecules is significant with respect to the processing time scales. The nonequilibrium thermodynamic theory of Durning and Tabor (1986) is applied to model 1-D drying of viscoelastic solutions with concentration-dependent physical properties. Transport of solvent to the surface of the coating occurs by viscoelastic diffusion down the gradient of a diffusion potential with a relaxing, nonequilibrium contribution. Galerkin's method with finite-element basis functions and a differential/algebraic equation system solver enable efficient solution of this stiff nonlinear model. Predictions show that elasticity enhances diffusion within the coating. At high Deborah numbers, however, a fall in the surface activity slows the rate of desorption. The coating thickness after a specified time under fixed total driving force is the smallest at intermediate Deborah numbers, showing that a small amount of viscoelasticity actually aids in drying. This can be interpreted as a skinning effect.