Pick my cotton in other words means 'do me a favor' but is not a question, it has more of a 'do it' tone.
Usually the favor being asked is a simple physical favor such as turning off a light or plugging in cables.
Alternatively it can be used as 'I'll pick your cotton' which is used to say, 'Yes I will do you a favor, but I hate you.'
"Grab my coffee while you're in the kitchen."
"Fine, I'll pick your cotton. Bastard."