In practice, many mechanical systems are underactuated, such as naval vessels, cranes, and helicopters, to reduce energy consumption and enhance flexibility. However, compounded by strong nonlinearity arising from state coupling, the underactuated nature and high-order unavailable states pose great challenges to motion control (particularly for unactuated states lacking independent actuators or kinematic constraints). In this paper, an adaptive controller based on fully-actuated system methods is proposed, together with a general and extensible analysis method. Firstly, a group of high-order auxiliary variables, consisting of actuated/unactuated states, their derivatives, and proportional-differential terms, are designed to rearrange the nonlinear underactuated system as a high-order linear fullyactuated system without any linearization operations. The asymptotic convergence of auxiliary variables theoretically eliminates the steady-state errors of actuated/unactuated states together. For high-order unmeasurable variables, they are recovered by the constructed neural network observer to estimate high-order dynamics, which avoids discontinuous robust terms and improves the accuracy of compensation/positioning. Motivated by the inherent features and advantages of fully-actuated systems, this paper proposes the first fully-actuated system-based continuous adaptive controller for a class of underactuated robots. Moreover, it is convenient to extend the proposed controller to handle more practical problems, such as state delay, without the need to re-conduct Lyapunov-based analysis. In addition to complete theoretical frames, this paper also provides several experimental validation.Remark:  This paper was first submitted to an IEEE Transactions journal on Mar. 28, 2024. It has been revised and submitted on Oct. 02, 2024.  Currently, this paper is under review.