Soft actuators are critical for enabling soft robots, medical devices, and haptic systems. Many soft actuators, however, require power to hold a configuration and rely on hard circuitry for control, limiting their potential applications. In this work, the first soft electromagnetic system is demonstrated for externally-controlled bistable actuation or self-regulated astable oscillation. This novel bellows-shaped actuator uses liquid metal encased in silicone as a compliant conductor that is capable of force generation, integrated sensing, and self-reconnecting. In the bistable configuration, the actuator can hold positions with no power. By utilizing a unique soft kinking mechanism, the actuator can generate feedback for self-regulated oscillation. The construction, sensing, and feedback mechanisms for this actuator are first discussed. Then, the force output, thermal performance, and dynamics are characterized. The bistable version has a stroke of 6 mm and can compress/expand with only 15 W of power for 30 ms. The astable version has a stroke of 3 mm and can oscillate at 27 Hz with 18 W of power. Several applications are demonstrated including bistable crawling, hopping, pulsing, and swimming. By adding a 20 V battery, self-regulated astable vibrational locomotion is also demonstrated. Overall, this work shows how these actuators and oscillators can bridge the gap between conventional and soft robots.