This paper investigates the Flying Sidekick Traveling Salesman Problem (FSTSP), and proposes an efficient Adaptive Large Neighborhood Search (ALNS) algorithm. Our proposed framework operates directly on a complete solution representation, eliminating the reconstruction step required by indirect encodings and making temporal information immediately accessible during the search process. A stage-based mechanism is introduced to efficiently update drone sorties while preserving the temporal consistency of drone operations when modifications are made to the truck route. In addition, destroy and repair operators are organized into coherent operator groups to avoid inconsistent intermediate solutions and improve large neighborhood exploration. The proposed method further incorporates six problem-specific local search moves and derives constant-time move evaluation formulas, with local search supported by an adaptation of the Static Move Descriptor (SMD) technique to accelerate the computational process. Experiments demonstrate that the proposed ALNS outperforms several state-of-the-art algorithms for the FSTSP, while maintaining stable performance on large-scale instances with up to 500 customers within 1800 seconds.