Aerial manipulation requires force-aware capabilities to enable safe and effective grasping and physical interaction. Previous works often rely on heavy, expensive force sensors unsuitable for typical quadrotor platforms, or perform grasping without force feedback, risking damage to fragile objects. To address these limitations, we propose a novel force-aware grasping framework incorporating six low-cost, sensitive skin-like tactile sensors. We introduce a magnetic-based tactile sensing module that provides high-precision three-dimensional force measurements. We eliminate geomagnetic interference through a reference Hall sensor and simplify the calibration process compared to previous work. The proposed framework enables precise force-aware grasping control, allowing safe manipulation of fragile objects and real-time weight measurement of grasped items. The system is validated through comprehensive real-world experiments, including balloon grasping, dynamic load variation tests, and ablation studies, demonstrating its effectiveness in various aerial manipulation scenarios. Our approach achieves fully onboard operation without external motion capture systems, significantly enhancing the practicality of force-sensitive aerial manipulation. The supplementary video is available at: https://www.youtube.com/watch?v=mbcZkrJEf1I.