where did the word 'skunt,' meaning to cut your grass very low, come from?

It's possible that this is a slang outgrowth of skinned. To skin something is to make it bare or remove the surface of something, which could be figuratively extended to cutting a short crop or lawn.

skinned > skint > skunt

Some sources I found in newspaper corpora use "skunt" to mean, clearly, "skinned."

he's dead an' gone to the Bad Place 'cause he skunt a cat--I don't mean skin the cat on an actin' pole like me an' Wilkes Booth Lincoln does--he skunt a sho' 'nough cat that was a black cat

• 1912 - Frances Boyd Calhoun, Dixon Evening Telegraph (Dixon, IL) 13 Jun, 6/1 (paywall)

Green's Dictionary of Slang lists skint as a slang variant of "skinned" with a cross reference to "skunt out," although it does not directly address this particular sense of "skunt."

without money, out of funds; thus as v., to make penniless.

The etymology here is listed as Standard English "skinned."

It looks likely that this sense of "skint" meaning "penniless" as a variation of "skinned" is similar to "skunt" meaning "cut bare" or "cut very short."

In a 1947 short story anthologized in Twenty-one Texas Stories, a man and his nephew discover that the caretaker in whose hands they left an inherited Hill Country farm has likely been helping himself to the profits as well as a few head of cattle:

"Because me and you's done been skunt out of no tellin how much money,” Uncle Mark said.

The expression is not limited to Texas, but also known in AAVE, or at least Joel Chandler Harris's version of it in 1880:

"So I kin git some water ter clean you wid atter I done skunt you, Brer Rabbit.' "Please, sir, lemme go, Brer Wolf.'"

So we've got Brer Rabbit about to get skinned, a couple of guys who got skinned by an untrusty caretaker, and most likely, ya'll's skinned lawns in Dallas.

Those who used skunting had apparently lost any sense of the word being a past participle and just begun using skunt as a regular verb.

Skint, however, is originally a British take on skinned from ca. 1925 and remains a primarily British term. It should thus be seen as a parallel rather than related form.