<<three sheets to the wind = loose ropes (as in "jib sheets")

From Webster's online:

Etymology: Middle English shete, from Old English scyte; akin to Old English scEat edge, Old High German scOz flap, skirt Date: before 12th century
1 a : a broad piece of cloth; especially : BEDSHEET b : SAIL 1a(1)

Thus, not rope, but the sail itself. "Three sheets to the wind" would be lots of sail in the wind and equivalent to "going full speed ahead."

Or?