This paper presents a new feedback-accelerated Picard iteration method for solving long-term orbit-propagation problems and perturbed Lambert's problems. This method is developed by combining the collocation method and the variational iteration method over large time steps. The resulting iterative formulas are explicitly derived so that they can be directly adopted to solve problems in orbital mechanics. Several typical orbit regimes incorporating high-order gravity and air drag force are used to demonstrate the application of the proposed method in orbit propagation. Further, the feedback-accelerated Picard iteration method is used to solve perturbed orbit-transfer problems. The combination of it with a fish-scale-growing method successfully extends its convergence domain and provides a potential approach for solving long-duration two-point boundary-value problems in conservative systems. The numerical results show that the proposed method is highly precise and efficient.