In this paper, we study the solution of convex generalized Nash equilibrium problems (GNEP) with shared linear constraints, and propose a class of regularized augmented Lagrangian methods. The idea is to penalize the shared linear constraints into the augmented Lagrangian function of each player, so as to construct a convex Nash equilibrium subproblem (NEP). Under the strong monotonicity and Lipschitz continuity assumptions of pseudo-gradient, we prove the Fejér monotonicity of iterative points with respect to the set of solutions. Under the cocoercivity assumption of pseudo-gradient, the iterative scheme of the algorithm is equivalent to the forward-backward splitting algorithm for solving the zero of an operator. If one more correction step is added, the cocoercivity hypothesis of pseudo-gradient can be weakened to the Lipschitz continuity hypothesis. Some numerical examples are given to verify the effectiveness of the algorithm.