The extended porous medium equation (PME) is a degenerate nonlinear diffusion equation that effectively describes moisture dynamics with evaporation in non-woven fibrous sheets. We propose a new finite volume numerical model of the extended PME incorporating regularization of nonlinear degenerate terms, and apply it to test cases for verification of accuracy, stability, and versatility. One of the test cases considered is a new exact steady solution of the extended PME. We also examine a differential equation-based adaptive re-meshing technique for resolving sharp transitions of solution profiles that may be optionally incorporated into the procedure above. The computational results demonstrate satisfactory accuracy of the proposed numerical model, with reasonable reproduction of complicated moisture dynamics involving sharp transitions and divorce of supports.