The widespread adoption of Electric Vehicles (EVs) in the smart grid is transforming the traditional grid into a more complex system. EVs have the ability to both charge and discharge, acting as loads that draw high power and sources that inject energy back into the grid. Consequently, energy sharing and management within smart grid communities integrated with EVs have become interesting aspects to study in order to efficiently utilize this energy. However, most existing research focuses solely on energy sharing within single communities, utilizing homogeneous energy profiles and neglecting the potential of heterogeneous energy across multiple communities. EVs also possess the capability to travel to different places and communities, where they can engage in energy sharing with areas that have varying load profiles and prices. In this work, a novel three-level energy sharing management approach is proposed for a multiple community system integrating movable energy storage such as EVs. This model involves three main entities: the Utility Company (UC), Community Energy Aggregator (CEA), and EVs. The energy sharing problem is formulated as a Stackelberg game, with all entities striving to maximize their utility through optimal strategies, including pricing, energy demand, or supply. The proposed model is validated through comparison with typical human charging behavior, as well as single- and multiple-community two-level game models. The findings reveal that the proposed model successfully optimizes pricing and energy strategies, significantly lowering the peak-to-average ratio and smoothing the overall energy profile.