In atypical industrial settings, soft grippers needed to adjust to different object shapes. Existing grabbers typically accommodated only single-curvature, fixed-stiffness objects, restricting their stability and usability. This study presents a design for a finger featuring multi-curvature, incorporating a wedge actuator alongside two variable stiffness units (VSUs) inspired by snake scales. By adjusting the high stiffness and low stiffness states of the variable stiffness element, the local structural stiffness of the finger was changed, thereby granting the gripper capabilities in bending shape control and variable stiffness. A finite element model of the wedge actuator was developed, and the influence of several parameters, including top wall thickness, side wall thickness, transition layer thickness, and sidewall height on bending angle and tip output force was analyzed through an orthogonal experiment. Furthermore, the relationship between the longitudinal length of the wedge actuator and both the bending angle and the tip output force was studied. Via explicit dynamic analysis, the stiffness variation of the VSU under operational vacuum pressure was predicted and subsequently validated against experimental data, confirming the reliability of the model. The effectiveness of finger shape control and stiffness adjustment was evaluated through experiments. Ultimately, a two-finger gripper was constructed to carry out the grasping experiments. The results showed that the gripper is capable of generating various clamping curvatures, enabling it to conform closely to the objects it grips and significantly broaden its clamping range.