Soft grippers have gained significant attention in robotics for their ability to adapt to diverse object geometries, comply with variable surfaces, and handle fragile items safely. Incorporating variable stiffness mechanisms has enhanced their performance by enabling adjustments in gripping force and structural rigidity. However, many existing designs primarily focus on stiffness modulation from the perspective of gripping force, limiting their ability to achieve localized control or handle objects with varying sizes, shapes, and weights effectively. This letter introduces a novel soft gripper integrating a movable variable stiffness mechanism and embedded soft sensors to overcome these challenges. The gripper locally modulates stiffness, allowing control over gripping force, contact area, and load distribution. Embedded soft sensors provide real-time feedback on gripping status, enabling stiffness adjustment and secure object handling. Fabricated as a monolithic structure through multi-material 3D printing, the gripper integrates functional components into a single manufacturing process with minimal post-processing. Experimental results validate the gripper's adaptability and versatility in handling delicate, irregular, and heavy objects, highlighting its potential for advanced robotic applications.