Grid-forming static var generators (SVGs) possess voltage-source characteristics that can enhance the stability and proactive voltage support capabilities of weak grids. The SVG with partial energy storage can provide inertia support. To address the problem that an energy-storage static var generator (ESVG) is prone to overcurrent when employing grid-forming control, this paper analyzes the joint control mechanism between the cascaded H-bridge and the energy storage system, based on the ESVG topology. Building on the current limiting mechanism and existing limiting strategies of the grid-forming ESVG, an adaptive virtual admittance current limiting strategy suitable for symmetrical faults is proposed. The strategy not only maintains the voltage-source characteristics of the ESVG but also significantly reduces transient overcurrent and overvoltage. Additionally, the strategy incorporates the clearing of transient virtual impedance to ensure its reliability. The effect of the virtual impedance ratio on stability is analyzed by a current closed-loop controller. Finally, the correctness of the theoretical analysis and the effectiveness of the proposed current limiting strategy are verified by simulation.