"Walk the chalk" might be a reference to "walking the chalk line" which I read in a book on Black dance {title?] referred to the practice of enslaved African American dancing or walking in a straight line with a 'glass' of water on their head..The one who didn't spill any or who spilled the least was the 'winner'.

If I remember correctly, "walking the chalk line" was a precusor to the doing the cakewalk.

See this reference to "walk chalk" in the second verse of "Gooseberry Wine'from Talley's "Negro Folk Rhymes":

"Oh walk chalk, Ginger Blue!
Git over double trouble.
You needn' min' de wedder
So d win' don't blow you double."
Talley, "Negro Folk Rhymes", p. 1}

-snip-

"Getting over double trouble" is a floating phrase that can be found in a number of folk rhymes. I interprete this verse as an exhortation from one Black person to another {or to Black people in general} to be extra careful. Regardless of the weather {what ever comes their way}..they must walk a chalk line {walk as carefully as they would walk doing the chalk line dance}.

IMO, "Ginger Blue" is probably a referent for a gingered colored Black person. This referent is similar to or the same as what some Black people still call a "Red bone" i.e. a Black person with reddish hues to his or her skin color.

And, off topic to this thread's title- IMO, the word "Pink" in Talley's "My pretty little Pink" is also a skin color referent
[in this case, a 'high yellow" African American.]

See the rhyme "My Pretty Little Pink":
"My pretty little Pink,
I once did think,
Dat we-uns shh',would marry;
But I's done give up,
Hain't got no hope,
I hain't got no time to tarry.
I'll drink coffe dat flows,
From oaks dats grows.
'Long se rive dar flows with brandy.

{Talley, "Negro Folk Rhymes", p. 127}