To address the dynamic changes in urgency and distinct emergency response phases in disaster relief, a multi-stage emergency material dispatch optimization model considering time windows and demand priorities is proposed. The rescue process is divided into three stages: Stage 1 represents the golden rescue period, aiming to minimize response time; Stage 2 corresponds to the mid-rescue phase, incorporating priority coefficients and soft time windows to construct a dual objective that balances timeliness and satisfaction; Stage 3 represents the late-rescue phase, incorporating supply point allocation to minimize overall logistics costs. To efficiently solve this multi-stage combinatorial optimization problem, a dual adaptive genetic simulated annealing algorithm (DA-GASA) is designed. Within the genetic algorithm framework, adaptive crossover and mutation probabilities are employed in the outer layer to maintain population diversity, while a simulated annealing operator based on variable neighborhood search is embedded in the inner layer to enhance local search and escape capabilities. Finally, numerical simulations based on the Wenchuan earthquake case are performed and compared with benchmark algorithms. The results indicate that the proposed model effectively characterizes the phased features of emergency dispatch, and the DA-GASA algorithm outperforms the compared algorithms in both convergence speed and solution accuracy, thereby providing a scientific reference for emergency decision-making.