Excessive noise in amplifier circuit when switching from DC power supply to AC-DC adapter
Based on your description, I am assuming your schematic should be reading +12v and 0v, rather than +/-12v which, as mentioned above, would yield a total of 24v.

I am also going to assume that the noise is a buzz or a hum, rather than a hissing, or white noise.

If that is the case, the first thing I would suspect is that the wall adapters are switching power supplies, and your adjustable supply is linear. From my understanding, certain amplifier circuits can be more sensitive to the noise generated by the switching action, whereas linear supplies do not produce such noise. So, I believe you may also be amplifying the noise generated by your adapter.