In disaster relief efforts, numerous challenges such as damaged roads and regional isolation render traditional transport vehicles like trucks and helicopters impractical. Consequently, alternative modes of transportation become imperative. Drones demonstrate their unique advantages in emergency rescue, garnering widespread attention. They possess high mobility and are not limited by ground traffic restrictions, enabling them to safely overcome obstacles and effectively prevent direct contact between the disastrous areas and external personnel, thereby reducing potential safety risks. However, the delivery capability of drones is limited by their battery capacity and payload, coupled with the pressure to achieve efficient delivery within urgent time windows. Particularly for items such as medical supplies and fresh produce, which require a high degree of timeliness, it is crucial to ensure that they retain the necessary freshness upon arrival at their destination. In order to meet the timeliness target for the delivery of disaster relief supplies and to improve the response efficiency of emergency relief, this paper proposes a drone delivery problem in emergency relief that considers energy constraints. Drones take off from starting points with rescue supplies and deliver along predetermined routes. During this process, the drone’s battery power gradually depletes due to the burden of the load and the duration of the flight. Each demand point is equipped with battery swapping devices. The battery swapping decisions for drones are dynamically adjusted based on their remaining power and delivery response level, to ensure the continuous operation of the drones. Since each battery swapping operation takes a certain amount of time, it is necessary to integrate the timeliness of the items and the time window constraints at demand points to enhance battery utilization, shorten the delivery time of supplies, and increase the delivery efficiency of drones in emergency rescue scenarios. The first section of the manuscript introduces the application scenarios for drone delivery in the emergency relief operations, defines the relevant parameters and decision variables, and constructs a mixed-integer linear programming model. This model comprehensively considers the payload capacity, battery capacity, timeliness, and the constraints of delivery time windows for drones, making battery swapping decisions based on the actual remaining battery life to enhance the response efficiency of the emergency relief operations. Given that the proposed model is NP-hard in nature, it is difficult for the commercial solvers to obtain optimal solutions within polynomial time for large-scale delivery demands. Consequently, the second section of this paper discusses in more details about the design of a branch-and-price exact algorithm within a column generation framework. This algorithm decomposes the original problem into a master problem and a pricing subproblem. To enhance the resolution of the subproblem, a strategy combining heuristic and a dynamic programming-based label-setting algorithm are implemented. The heuristic algorithm swiftly generates initial feasible paths, whereas the label-setting algorithm seeks the shortest possible routes under resource constraints, ensuring the optimality of solutions through dominance rules. Furthermore, to address the challenges of the large-scale drone delivery, a metaheuristic algorithm based on large neighborhood search is designed, ensuring that practical and implementable delivery solutions are provided in a short time. In the third section, a series of numerical experiments are presented to validate the effectiveness of the proposed optimization model and algorithms. The empirical findings demonstrate that the designed branch-and-price algorithm is capable of identifying optimal delivery solutions that meet the timeliness target in a short time. Compared to the heuristic algorithm, the branch-and-price algorithm reduces the number of drones required, thereby lowering operational costs. Additionally, the dynamic battery swapping strategy proposed in this paper significantly reduces the frequency of battery swapping, consequently elevating the efficiency of delivering rescue supplies. Overall, the emergency drone delivery model designed in this paper effectively utilizes drone characteristics and considers the strict requirements for time windows and freshness of supplies in actual rescue scenarios, providing an effective optimization model and algorithms for drone delivery in emergency situations.