Liquid shells (e.g., double emulsions, vesicles, etc.) are susceptible to interfacial instability and rupturing when driven out of mechanical equilibrium. This poses a significant challenge for the design of liquid-shell-based micromachines, where the goal is to maintain stability and dynamical control in combination with motility. Here, we present our solution to this problem with controllable self-propelling liquid shells, which we have stabilized using the soft topological constraints imposed by a nematogen oil. We demonstrate, through experiments and simulations, that anisotropic elasticity can counterbalance the destabilizing effect of viscous drag induced by shell motility and inhibit rupturing. We analyze their propulsion dynamics and identify a peculiar meandering behavior driven by a combination of topological and chemical spontaneously broken symmetries. Based on our understanding of these symmetry breaking mechanisms, we provide routes to control shell motion via topology, chemical signaling, and hydrodynamic interactions.