To mitigate the long-term spectrum crunch problem, the FCC recently opened up the 6 GHz frequency band for unlicensed use. However, the existing spectrum sharing strategies cannot support the operation of access points in moving vehicles such as cars and UAVs. This is primarily because of the directionality-based spectrum sharing among the incumbent systems in this band and the high mobility of the moving vehicles, which together make it challenging to control the cross-system interference. In this paper, we propose SwarmShare, a mobility-resilient spectrum sharing framework for swarm UAV networking in the 6 GHz band. We first present a mathematical formulation of the SwarmShare problem, where the objective is to maximize the spectral efficiency of the UAV network by jointly controlling the flight and transmission power of the UAVs and their association with the ground users, under the interference constraints of the incumbent system. We find that there are no closed-form mathematical models that can be used to characterize the statistical behaviors of the aggregate interference from the UAVs to the incumbent system. Then we propose a data-driven three-phase spectrum sharing approach, including Initial Power Enforcement, Offline-dataset Guided Online Power Adaptation, and Reinforcement Learning-based UAV Optimization. We validate the effectiveness of SwarmShare through an extensive simulation campaign. Results indicate that, based on SwarmShare, the aggregate interference from the UAVs to the incumbent system can be effectively kept below the target level without requiring the real-time cross-system channel state information. The mobility resilience of SwarmShare is also validated in coexisting networks with no precise UAV location information.