Soft robotic end-effectors, offering superior adaptability, safety, and flexibility, enable robots to perform complex and delicate tasks. However, passive adhesion often results in insufficient contact with target surfaces, limiting grasping stability and locomotion. To address this issue, we present a shape memory alloy (SMA)-driven electrostatic adhesive soft end-effector (SEAE) that integrates deformation actuation with electrostatic adhesion control. The SEAE comprises an SMA-actuated module and an electrostatic adhesive module. By selectively heating two embedded SMA wire layers within the actuation module, the electrostatic adhesive unit at the end of the SEAE is cyclically driven to achieve dynamic adhesion and detachment from target surfaces. Based on this design, a wall-climbing robot, an inchworm-like crawling robot, and a soft gripper are developed using SEAEs as their feet or fingers. Experimental results validate the feasibility of the proposed SEAE concept for climbing, crawling, and grasping tasks across representative surface materials and geometries. The SEAE concept proposed here demonstrates the feasibility of integrating electrostatic adhesion into soft robotic systems and suggests its potential applicability in a broader range of soft robotic scenarios.