Robots-based smart pharmacies are essential for modern healthcare systems, enabling efficient drug delivery. However, a critical challenge exists in the robotic handling of drugs with varying shapes and overlapping positions, which previous studies have not adequately addressed. To enhance the robotic arm's ability to grasp chaotic, overlapping, and variously shaped drugs, this paper proposed a novel framework combining a multi-stage grasping network with an adaptive robotics mechanism. The framework first preprocessed images using an improved Super-Resolution Convolutional Neural Network (SRCNN) algorithm, and then employed the proposed YOLOv5+E-A-SPPFCSPC+BIFPNC (YOLO-EASB) instance segmentation algorithm for precise drug segmentation. The most suitable drugs for grasping can be determined by assessing the completeness of the segmentation masks. Then, these segmented drugs were processed by our improved Adaptive Feature Fusion and Grasp-Aware Network (IAFFGA-Net) with the optimized loss function, which ensures accurate picking actions even in complex environments. To control the robot grasping, a time-optimal robotic arm trajectory planning algorithm that combines an improved ant colony algorithm with 3-5-3 interpolation was developed, further improving efficiency while ensuring smooth trajectories. Finally, this system was implemented and validated within an adaptive collaborative robot setup, which dynamically adjusts to different production environments and task requirements. Experimental results demonstrate the superiority of our multi-stage grasping network in optimizing smart pharmacy operations, while also showcasing its remarkable adaptability and effectiveness in practical applications.