Coupling-Tiltable Unmanned Aerial-Aquatic Vehicles (UAAVs) have gained increasing importance, yet lack comprehensive analysis and suitable controllers. This paper analyzes the underwater motion characteristics of a self-designed UAAV, Mirs-Alioth, and designs a controller for it. The effectiveness of the controller is validated through experiments. The singularities of Mirs-Alioth are derived as Singular Thrust Tilt Angle (STTA), which serve as an essential tool for an analysis of its underwater motion characteristics. The analysis reveals several key factors for designing the controller. These include the need for logic switching, using a Nussbaum function to compensate control direction uncertainty in the auxiliary channel, and employing an auxiliary controller to mitigate coupling effects. Based on these key points, a control scheme is designed. It consists of a controller that regulates the thrust tilt angle to the singular value, an auxiliary controller incorporating a Saturated Nussbaum function, and a logic switch. Eventually, two sets of experiments are conducted to validate the effectiveness of the controller and demonstrate the necessity of the Nussbaum function.