https://en.wikipedia.org/wiki/United_States_Declaration_of_Independence 
 has a good related article which may explain the origin of this claim: Many "declarations" were resolutions adopted at town or county meetings that offered support for independence. A few came in the form of jury instructions, such as the statement issued on April 23, 1776, by Chief Justice William Henry Drayton of South Carolina: "the law of the land authorizes me to declare...that George the Third, King of Great Britain...has no authority over us, and we owe no obedience to him."[38] Or maybe there is some confusion with SC being the first to secede from the Union. 
 A Google search for "south+carolina+first+state+to+declare+independence+American+revolution" shows nothing consistent with the claim other than this South Carolina article. It seems reasonable to remove it until a solid reference is provided. 

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