This work proposes a nonsingular adaptive fixed-time switching control method for a class of strict-feedback nonlinear dynamics subject to full state constraints. The peculiarity of this design lies in overcoming the singularity issue that typically appears in the existing backstepping-based fixed-time control methods caused by the iterative differentiation of fractional power terms as tracking errors approach to zero, while guaranteeing the nonviolation of full state constraints. Crucial in solving such singularity issue is to skillfully introduce a smooth switching between fractional power and integer power terms, which guarantees that fractional power term is confined within a positive interval all the time. An asymmetric time-varying barrier Lyapunov function is delicately incorporated into control design, rendering state variables to be within prescribed time-varying bounds. Besides, radial basis function neural network is employed to handle system unknown nonlinearities. It is rigorously proved that all the closed-loop signals eventually converge to small regions around origin within fixed-time. Comparative simulation results are finally given to validate the effectiveness and superiority of the proposed control strategy.