Task offloading in edge computing becomes significantly more challenging when applications are modeled as Directed Acyclic Graphs (DAGs), where task dependencies exist within each instance and extend across consecutive instances, chaining them over time in a mobile environment. In addition, each task requires a specific service, but edge servers can provide only a limited subset of services. As the spatial distribution of service demand changes with user movement, the cached services must be updated accordingly, which in turn affects the offloading decisions that depend on service availability. This paper proposes a dependency-aware representation learning framework for task offloading in such environments. A Transformer encoder is pretrained offline using structural supervision signals derived from DAG properties, such as critical-path workload and slack, to capture task importance and dependency structure. The learned embeddings are integrated with system-level features to construct the state of a Double Deep Q-Network (DDQN) agent deployed at each access gateway. Ablation studies show that task embedding and service availability are the most influential components in offloading decisions, and that their effectiveness depends on complementary system information. Simulation results demonstrate that the proposed method consistently reduces application completion time across varying CPU frequencies, task workloads, network scales, and background server loads, outperforming DDQN baselines with handcrafted features and heuristic methods.