"I thought the 'Rye' referred to in Robert Burns' poem was the river Rye, hence the lines: 'Jenny's a wet poor body, Jenny's seldom dry'. In this regard it is about two people who meet at a river with no crossing, which will cause people to question why one of them is wet and what they have been doing."
No. Rye means a field of rye. Remember that this is a sexually-themed poem. When Burns says that "Jenny is rarely dry," he is referring to her vaginal lubrication. Jenny is sexually active, so her genitals are seldom dry.