I need to add a variable phase shift circuit to my Wien Oscillator, I have done some research, but have only found phase shift oscillators made with 3 RC stages that generate a 180º shift. However, I don't know how to add this with a Wien bridge, nor how to make it variable.
Your version is quite complex. Consider to use more classic one opamp version, which also is easier to make adjustable. See for example this:
(sorry I have no right to copy the images)
There's also the diode stabilized version.
At the oscillation frequency the non-inverting input gets the signali from the output zero degrees phase shifted. - the phase lead and lag of the RC circuit compensate each other. The gain of the opamp is made a little larger than the attenuation of the RC circuit. The diodes start to conduct at certain output amplitude which reduces the gain and stops the amplitude growth.
Replace the fixed resistors in the RC positive feedback circuit (= Wien Bridge) with a tandem potentiometer. The lower is the resistance, the higher is the oscillation frequency. That's also well presented in the linked article.
You must not exceed the output current capability nor the usable frequency range of the opamp.