Yes, and the amplitude changes with ambient temperature as well.
Nothing wrong with the Wien Bridge circuit, it gives extremely low distortion, lower than just about anything else, which is why it is used in many audio signal generators used for measuring distortion in audio equipment.
There are many methods to stabilise the output amplitude. Lamps and diodes are the simplest, but you could also use a FET or a light dependant resistor, or even an analog multiplier if you wanted.