The control and optimization of grid-forming converters continue to be vital research topics in power electronics. One fundamental task is limiting the converter's output current during severe grid disturbances to ensure equipment protection. A novel current-limiting approach is the backstepping design with sigmoid-shaped control Lyapunov functions, resulting in one continuous control law, applicable in both normal and fault conditions. This paper expands on the backstepping design for a two-level voltage source converter with LCL-filter with an innovative integrator design to guarantee stationary accuracy and rapid switching between operation regimes. In addition, a novel approach for adaptive current-limiting parameters is proposed, to avoid unwanted phase shifts during low voltage ride through. Measurements show the functionality and the effectiveness of the design under both nominal and fault conditions.