Pretty much every oscillating fan I've ever seen uses a shaded pole AC motor with tapped windings for speed control. That means that the speed of fan rotation is only controlled by the power the motors puts out, and that depends on which tap on the winding is, well, tapped. There is nothing electronic in the fan (unless this is some very high end model - can you post the part number?) that can control the speed in the fashion that you describe, which is why it is almost a guarantee the issue is mechanical and it is very often dust and hair that finds its way in and winds around any axle that's turning inside that fan.
There are parts of the oscillating mechanism gearbox that are always engaged to the motor and parts that you can disengage. If disengaging does not change the fan rotation behavior, specifically look for any problem - dirt, hair or perhaps even parts of the plastic body of the gear box, anything that does not belong - right around the motor shaft, the worm gear and the first gear engaged to the worm. You'll have to remove the cover of the fan and then remove another cover off of that plastic gear box that sits right behind the motor to get access to the worm gear.