We study the dynamics of two pinned droplets under the influence of an applied electric stress. We find that at a sufficiently strong field, this stress is sufficient to induce contact of the droplets. Interestingly, upon such contact, the dynamic behavior sensitively depends on the separation distance between the droplets. Besides the classical "coalescence" regime, we identify two other dynamic regimes: "fuse-and-split" and "periodic non-coalescence." In the "fuse-and-split" regime, the droplets first fuse to form a jet, which subsequently breaks up into two droplets. In the "periodic non-coalescence" regime, the droplets contact and bounce away periodically without coalescence. Further analysis indicates that while the electric stress stretches the droplets into shapes that depend on the initial droplet separation, the surface tension stress dominates over the electric stress as soon as the droplets touch. We show that the shapes of the contacting droplets determine their subsequent dynamics. Our work provides a rationale for understanding the interplay between surface tension and electric stresses that govern the behavior of charged droplets and could inspire new methods for characterizing emulsion stability and surfactant performance.