Does an AC-DC adapter draw more current on 110V than while plugged to a 220V supply?
Yes, a universal input adapter will keep its output voltage constant when powered from (e.g.) 100-240 VAC.  That means it's supplying the same output power.

Since input power is output power + losses, and power is voltage * current, the higher the input voltage the lower the input current.  This is called a constant power load and is typical of switching power supplies.