Beams with fast and continuously-tunable orbital angular momentum (OAM) have potential applications in classical and quantum optical communications, sensing, and in the study of beam propagation through turbulence. An acousto-optical deflector (AOD) is a sophisticated, well-studied device that continuously and rapidly tunes the deflection angle of an output beam. The log-polar HOBBIT setup can generate beams with OAM by wrapping elliptically shaped Gaussian beams with linear phase tilt to a ring. By combining the linear tilted output from the AOD with the OAM generation capabilities of the HOBBIT system, the generated OAM modes become continuously tunable at high speeds measured on the order of 400 kHz.