To achieve precise control of terminal states for near-space vehicles under underactuated conditions, it is necessary to overcome the limitations of onboard computational capability and develop a highly precise and efficient trajectory optimization framework in computational guidance. If the number of discretization points is too low, existing Newton-type trajectory optimization methods by deviation dynamic equations (Model Predictive Static Programming) have large terminal control errors. Motivated by current quasi-spectral Newton method, a quasi-spectral optimization method is proposed by actual dynamic equations. Unlike existing methods that update the optimization coefficient indirectly, this method updates the optimization coefficient directly. Additionally, it is proven that this method is a Newton optimization method. And it is also clarified that when damping mechanism is not considered, it suffers from poor robustness during multiple iterations. To address the poor robustness and low control accuracy in typical Newton methods, an adaptive trajectory optimization method is proposed with fuzzy control. This method avoids the divergence problem commonly encountered in damping-free methods with sensitive initial values, thereby enhancing both robustness and accuracy. Simulation results show that the proposed method achieves faster dynamic trajectory optimization than existing methods under identical conditions.