Soft robotic hands with integrated sensing capabilities hold great potential for interactive operations. Previous work has typically focused on integrating sensors with fingers. The palm, as a large and crucial contact region providing mechanical support and sensory feedback, remains underexplored due to the currently limited sensing density and interaction with the fingers. Here, we develop a sensorized robotic hand that integrates a high-density tactile palm, dexterous soft fingers, and cooperative palm-finger interaction strategies. The palm features a compact visual-tactile design to capture delicate contact information. The soft fingers are designed as fiber-reinforced pneumatic actuators, each providing two-segment motions for multimodal grasping. These features enable extensive palm-finger interactions, offering mutual benefits such as improved grasping stability, automatic exquisite surface reconstruction, and accurate object classification. We also develop palm-finger feedback strategies to enable dynamic tasks, including planar object pickup, continuous flaw detection, and grasping pose adjustment. Furthermore, our development, augmented by artificial intelligence, shows improved potential for human-robot collaboration. Our results suggest the promise of fusing rich palm tactile sensing with soft dexterous fingers for advanced interactive robotic operations.