Cotton canvas as shown in your example pillowcase should engrave ok. Cotton turns a nice brown color when you engrave it. You’ll want to keep the power pretty low and the speed high, to avoid burning through the fabric. Thinner cotton fabrics will burn through easily but canvas (e.g. cotton duck) or denim work pretty well.
Dyed cotton will often turn white even at power level 1 (or close to 1).
Settings vary a lot depending on the exact fabric. What’s too low for one fabric may leave you with a charred hole in another fabric. You should probably buy a spare pillowcase to use for testing. And if the fabric is too thin you may find that it’s impossible to avoid charring.