R2 and R6 are not the problem. The extra noise you get from using values that are higher than necessary here would just be hiss, and it would be constant. It would literally be physically impossible for it to change, because the resistor values would have to change.
I don't see that you've ruled out noise from the DAP. What with the high gain and higher fidelity of the new amp, you might be hearing noise that was already there, but inaudible. You could be hearing power supply noise from the hard drive, for example, with the 10 second delay being the time it takes for the player to power down the drive.
If you retry the current draw test, don't do it with music playing. If you have to play music to get the noise to start, go ahead and make it start, but then stop the music and wait for the noise to stop and see if the current draw falls on the noise stopping. What I'm getting at is that this 3mA jump you're seeing may be due to the varying power requirement of driving music through the headphones. We need to be sure, though.