Robot-to-human handovers often rely on static, open-loop strategies (or, at best, approaches that adapt only the position), which generally do not consider how the object will be grasped by the human, thus requiring the user to adapt. This work presents a novel adaptive framework that dynamically adjusts the object's delivery pose in real time based on the user's hand pose and the intended downstream task. By integrating AI-based hand pose estimation with smooth, kinematically constrained trajectories, the system ensures a safe approach and an optimal handover orientation. A comprehensive user study compares the proposed adaptive approach against a static baseline across multiple tasks, evaluating both subjective metrics (NASA-TLX, Human-Robot Trust Scale) and objective physiological data (blink rate measured via wearable eye-trackers). The results demonstrate that dynamic alignment significantly reduces users' cognitive workload and physiological stress, while increasing perceived trust in the robot's reliability. These findings highlight the potential of task- and pose-aware systems for enabling fluid and ergonomic human-robot collaboration.