"Sticky" does make a lot more sense, but even Lightfoot's own website has it as "sickly." IMO, it just shows to go ya that having a lyric printed on a liner note or sheet music or at the author's website does not necessarily mean it's the definitive lyric. (Ian Tyson's website has several lyric errors for his songs, too, at least not the words he sings on his recordings.)

Anyway, I'm gonna sing "sticky."

Here are the words per Lightfoot's website with the following changes (corrections):
--"Sticky" instead of "sickly" and "where" instead of "when" in the first line of the last verse (because "sickly" and "when" don't make sense.
--"And a valley below" in the first verse (as Anne Murray sings it), instead of "in the valley below."  (How can the house be on a hill and in the valley at the same time?)
 

Cotton Jenny
Written by Gordon Lightfoot.
(© Early Morning Music/SOCAN/ASCAP.)  From "Summer Side Of Life", © 1971, Reprise Records

There's a house on a hill,
By a worn-down weathered old mill,
And a valley below where the river winds.
There's no such thing as bad times.
And a soft southern flame,
Oh, Cotton Jenny's her name,
And she wakes me up when the sun goes down,
And the wheel of love goes 'round.

                                Wheels of love go 'round, love go 'round,
                                Love go 'round, a joyful sound.
                                I ain't got a penny for Cotton Jenny to spend, but then,
                                The wheels go 'round.

When the new day begins, I go down to the cotton gin,
And I make my time worthwhile for them;
Then I climb back up again.
And she waits by the door;
Oh, Cotton Jenny, I'm sore.
And she rubs my feet while the sun goes down,
And the wheel of love goes 'round.

                                Wheels of love go 'round, love go 'round,
                                Love go 'round, a joyful sound.
                                I ain't got a penny for Cotton Jenny to spend, but then,
                                The wheels go 'round.

In the hot, sticky  South, where they say: "Well, shut my mouth,"
I can never be free from the cotton grind,
But I know I got what's mine
With a soft southern flame,
Oh, Cotton Jenny's her name.
She wakes me up when the sun goes down,
And the wheel of love goes 'round.

                                Wheels of love go 'round, love go 'round,
                                Love go 'round, a joyful sound.
                                I ain't got a penny for Cotton Jenny to spend, but then,
                                The wheels go 'round.

                                The wheels go 'round.
                                Round and round.

 

Anne Murray's version omits the last verse, probably because she sings it as 3rd person narrative, and the last verse doesn't work that way.
Murray also sings "The wheels of love go 'round" as the last line of each verse and as the last line of the chorus.

There's a clip of Gordie singing this song at his website, and he definitely sings "The wheel of love goes 'round" as the last line of the verse.  The clip cuts off before he finishes the chorus, though, so I don't know if he finishes the chorus as "The wheels go 'round" or "The wheels of love go 'round."