Two sources of noise come to my mind:
1. Some battery charging circuit. I'm not sure how these work and can't explain why touching the screen helps.
2. CPU switching power states. Disconnecting power cable could help by causing the OS to avoid high power states, using the touchscreen could help by causing it to temporarily stay in high power state to increase UI responsiveness.
IOW, experiment with power management settings. IIRC, Windows has some "battery saving" function which disables charging when battery is above 70% full. Try it. Also, try tweaking CPU power/performance settings if there are any. See if loading all CPUs at 100% helps.
As for transformers, besides eliminating common-mode noise, their inductance may theoretically kill some noise AM-modulated on an ultrasonic carrier. Maybe that's why they help with SMPS whine, IF they really help. Nevermind, this doesn't make sense with headphones.