Hydrogels may exhibit diffusion-deformation coupling effects due to both chemical and mechanical loadings. Since hydrogels are composed of a crosslinked polymer network and solvent, that are both nearly incompressible, these constituents may result in a nearly incompressible aggregate at the early stages of the transient problem. This early stage response is not suitable for finite elements with equal-order linear interpolation for displacement and chemical potential due to the lack of the inf-sup condition and volumetric locking. In this work, an adaptively stabilized equal order mixed finite element scheme is developed to treat these deficiencies both for 2D and 3D problems. A general strategy that unifies the stabilization procedure and cures the volumetric locking for the nonlinear theory for hydrogels and linear poroelasticity is established. Particular attention is given to the differentiation of physical and numerical stability issues in the transient response of hydrogels. The proposed method is then verified through a series of numerical examples that identify the numerical issues encountered in each numerical example.