To remedy the issues of explosion of complexity, nonconvergent parameter estimation, control singularity, and stringent requirement on the full states in the conventional backstepping methods, this paper proposes a new adaptive identification scheme with guaranteed convergence and an adaptive output-feedback control method for nonlinear strict-feedback systems without using backstepping. By defining new state variables and a coordinate transformation, the state-feedback control of strict-feedback systems can be transformed into an output-feedback control of the derived canonical model. Then, an one-step adaptive control with high-order differentiators is designed. Furthermore, two neural networks (NNs) are employed to approximate the lumped unknown dynamics so as to avoid the potential singularity in the existing control designs. To ensure the convergence of parameter estimation, a novel adaptive law is constructed to online update the NN weights. We adopt the Lyapunov method to prove that the tracking error and estimation error can converge to a small compact set around zero. Comparative simulations based on a hydraulic servo system are performed to show the effectiveness and superiority of the proposed methods.