The nonlinear analytic nodal method, which is formulated by combining the nonlinear iteration technique and the analytic nodal method (ANM), requires analytic solutions of the two-node problems. When the method is applied to problems that contain near-critical nodes in which there is essentially no net leakage, the two-node ANM solution for such nodes results in highly ill-conditioned matrices and potential numerical instabilities, especially in single precision arithmetic. Two stabilization techniques are introduced to resolve the instability problem by employing alternate basis functions for near-critical nodes. The first uses the exact ANM solution for a critical node, and the second employs the nodal expansion method. Both techniques are shown to perform well; however, the solution accuracy can be mildly sensitive to the criterion used to invoke the stabilized coupling kernel.