Action-stability problems for properly-degenerate Hamiltonian systems with two degrees of freedom allowing the intermediate system depending explicitly upon the angle-variable are considered. Under suitable ``convexity'' requirements for the intermediate Hamiltonian it is proved that the action variables stay forever close to their initial values in every energy surface. In more complex ``nonconvex'' cases this stability is conserved up to a small set, where the degenerate action-variable might drift away from its initial value by a quantity independent of the perturbation. Analysis near separatrices, KAM techniques and energy conservation arguments are used in proofs. In appendices the D'Alembert planetary model is considered and some details for the construction of the real-analytic action-angle variables for the pendulum are provided.