We investigate the tracking control problem for the class of the non-strict feedback nonlinear system with external disturbance and time-varying asymmetric full state constraints. Firstly, the unknown nonlinear term with external disturbance in the system are estimated by fuzzy logic system. The backstepping method is applied to the design of adaptive fuzzy controller. However, to prevent that the constraints are overstepped. By introducing novel tan-type time-varying asymmetric barrier Lyapunov function (TABLF) in each step of backstepping design. Secondly, the dynamic surface control (DSC) is introduced in the algorithm design to solve the complex problem of controller caused by the derivative of virtual control value. The proposed control scheme can speed up the tracking speed of the system. Compared with the previous work, it is verifified that the combination of DSC and TABLF can obtain good performance within the constraint range, and can ensure fast and stable tracking convergence under external disturbance. Finally, two simulation examples verify the performance of the adaptive controller.