Problems with low-side MOSFET in 12V LED dimming switch
However the LED sometimes randomly flickers at a very low brightness
in the dark even if it has no contacts to any conductive objects.

Your eyes are very sensitive to low light, and LEDs can work on very little current.
Simplest solution is to wire a resistor across the LED strip which shunts away any small currents that might be induced in the wiring. A 1 kΩ resistor would only increase current draw by 12 mA when the LEDs are on, and keep them off at lower induced currents.

The brightness increases when I touch the tab of the MOSFET or the GND
line of the LED

Then don't do that. Problem solved!