Previously reported acoustic metasurfaces consisting of fixed channels are untunable to meet broadband requirements and alterable functionalities. To overcome this limitation, we propose a screw-and-nut mechanism of tunability and design a type of continuously tunable acoustic metasurface with unit components of helical cylinders which are screwed into a plate. The spiral channel length can be tuned continuously by the screwed depth, and thus a metasurface with a continuously tunable acoustic phase of independent pixels is attained. Different distributions of the unit components can form an arbitrary metasurface profile. We also develop an approximate equivalent medium model to predict the tunability of the unit component. As an example, we present the design details of a circular tunable metasurface for three-dimensional (3D) adjustable broadband acoustic focusing of different airborne sound sources in a wide frequency range. A sample of the metasurface is manufactured by polylactic acid plastic with 3D-printed helical cylinders. Experiments of the adjustable acoustic focusing are performed to validate the equivalent medium model and the finite element simulation. It is shown that the results of the equivalent medium model, the finite element simulation, and the experiments are in good agreement.