How does changing arduino power supply from USB to Adapter change the circuit?
I have run into problems like this before, often it can actually be that the voltage is not steady due to the LEDs sourcing current from it.

You may be able to solve the problem by adding a capacitor inline with the power input (from the +12v to GND). I would probably start with a value of around 10 &mu;F and increase it if the interference continues. 

Capacitive sensors and very sensitive to voltage fluctuations, you may even want to put a second capacitor across the VCC and Gnd of the sensor.