I too have experienced noise from cheap switch mode wall warts. I did make them go away with a fairly humble ground loop isolator but I gave that away to another littleBits user so I'm stuck on battery power for now.
I have an old fashioned transformer based power supply (if it's heavy it's a real transformer) and that works cleanly. It claims 12v but multimeter shows about 18v (unloaded) so I need to check what's really coming out of that, e.g. could be rectified only with no smoothing. Power module is marked 9-12 so I don't want to cook anything.
Perhaps you could accept the noise whilst noodling and then when you want to record it switch to battery briefly? That would avoid wasting battery. Or use rechargeable?
BTW, I found when the battery starts going flat the oscillators tuning changes so watch out for that too.