Electric carsharing, i.e., electric vehicle sharing, as an emerging mobility-on-demand service, has been proliferating worldwide recently. Though providing convenient, low-cost, and environmentally-friendly mobility, there are also some potential roadblocks in electric carsharing services due to existing inefficient fleet management strategies, which relocate the vehicles using predefined periodic schedules without self-adapting to the highly dynamic user demand, and many practical factors like time-variant charging pricing also have not been fully considered. To remedy these problems, in this paper, we design Record, an effective fleet management system with joint Repositioning and Charging for electric carsharing based on dynamic deadlines to improve its operating profits and also satisfy users' real-time pickup and return demand. Record considers not only the highly dynamic user demand for vehicle repositioning (i.e., where to relocate) but also the time-varying charging pricing for charging scheduling (i.e., where to charge). To perform the two tasks efficiently, in Record, we design a dynamic deadline-based distributed deep reinforcement learning algorithm, which generates dynamic deadlines via usage prediction combined with an error compensation mechanism to adaptively search and learn the optimal locations for satisfying highly dynamic and unbalanced user demand in real time. We implement and evaluate the Record system with 10-month real-world electric carsharing data, and the extensive experimental results show that our Record effectively reduces 25.8% of charging costs and reduces 30.2% of vehicle movements by workers, and it also satisfies user demand and achieves a small runtime overhead at the same time.