We propose a dynamic term-by-term VMS-stabilized three-field mixed finite element formulation for generalized Newtonian fluids [1]. The stabilized formulation allows equal-order interpolations between the unknown fields and the numerical modeling of convective dominant flows of shear-thinning and shear-tickening fluids. Motivated by non-Newtonian constitutive models involving nonlinearities in the apparent viscosity, we consider a local postprocessing scheme to obtain a superconvergent postprocessed approximation of the velocity, which is computed (locally and in parallel) from the velocity and the stress approximations. We also propose and analyze a new post-processed approximation based on local minimization problems, whose objective function is the norm of the residual associated with the local media in a dual norm on an enlarged (discrete) test space. An equivalent rewriting of the minimization problem, as a saddle point problem, yields a dual variable whose norm can be used as an a posteriori error indicator. We prove that an indicator based on the postprocessed approximation and the dual variable is reliable and efficient for a norm based on the postprocessed velocity approximation [2]. As an additional advantage, we can feed the nonlinear loop with the postprocessed discrete velocity, which is a better approximation than the original one. We provide numerical evidence of the performance of the a posteriori error indicator and the postprocessed velocity approximation.