Your guess about the sensor principle is correct: When torque is applied to any shaft, the surface of the shaft gets stressed and strained. In directions of +-45° to the axis, there is pure compressive strain in one of the directions and pure tensile strain in the other.
The cuts in the foil of this sensor provide a preferred direction for the magnetic field and make both halfs a differential arrangement: When the impedance (inductance) increases in one half, it decreases in the other. The reason for the change in impedance cannot be told just from the picture, especially not without a look at the coils. It could be a geometric effect or a variation of foil permeability depending to strain, or both.
Circuit-wise, the differential arrangement of the coils translates to a measurement circuit like a differential amplifier, or a half-bridge, if you like, with two constant and two variing impedances.
The strain must be small, because any measurement system must avoid getting into the region of plastic deformation, which results in permanent property changes. For anything made from metal, this dictates tensile strain <<1%. In consequence, the signal also shows very little modulation of the stimulus signal. Expected behaviour! Same for the offset you get, i.e. output signal not zero without torque: These devices struggle to have or maintain the symmetry needed to avoid a large offset.
For optimization, you can:
- Test for the most effective frequency, if you are free to set it. Torque signal will increase with frequency, up to the point where losses or the low-pass characteristic of the foil prevent more gain.
- Increase amplitude of stimulus signal, depending on the maximum current the devices can bear or the maximum power you have for the job. Possible problem: self-heating.
Apart from this, yes, a differential amplifier with high gain and good CMRR is what you need.
I do not understand what sinusoidal DC is, so I'm not sure about this. If you mean a sine with a DC component added: That DC component would, in 1st order approximation, not change the torque signal, because it would result in a DC component of the output. But your torque information is, in telecommunication terms, the modulation of the carrier (=stimulus above). Adding DC just wastes power and heats up the sensor. If both is non-critical, why not.
One more thing: With too much current, whether AC or DC, it can also happen that the foil gets magnetically saturated, resulting in the torque signal to derease or even disappear. This sets another limit to the current, and suggests not to load the sensor with DC for optimum performance. Instead, using all 'current budget' for AC will result in best torque gain. It should be possible to detect saturation by comparing input and output waveforms: Since the sensor should be linear below saturation, both should be the same.