The South Carolina link was a separate document - the Declaration of Immediate Causes.

https://web.archive.org/web/20040404171724/http://members.aol.com/jfepperson/ordnces.html

South Carolina’s ordinance of secession is pretty boilerplate

AN ORDINANCE to dissolve the union between the State of South Carolina and other States united with her under the compact entitled “The Constitution of the United States of America.”

We, the people of the State of South Carolina, in convention assembled, do declare and ordain, and it is hereby declared and ordained, That the ordinance adopted by us in convention on the twenty-third day of May, in the year of our Lord one thousand seven hundred and eighty-eight, whereby the Constitution of the United States of America was ratified, and also all acts and parts of acts of the General Assembly of this State ratifying amendments of the said Constitution, are hereby repealed; and that the union now subsisting between South Carolina and other States, under the name of the “United States of America,” is hereby dissolved.

Done at Charleston the twentieth day of December, in the year of our Lord one thousand eight hundred and sixty.

… while Virginia’s cites slavery specifically in its ordinance.

AN ORDINANCE to repeal the ratification of the Constitution of the United State of America by the State of Virginia, and to resume all the rights and powers granted under said Constitution.

The people of Virginia in their ratification of the Constitution of the United States of America, adopted by them in convention on the twenty-fifth day of June, in the year of our Lord one thousand seven hundred and eighty-eight, having declared that the powers granted under said Constitition were derived from the people of the United States and might be resumed whensoever the same should be perverted to their injury and oppression, and the Federal Government having perverted said powers not only to the injury of the people of Virginia, but to the oppression of the Southern slave-holding States:

Now, therefore, we, the people of Virginia, do declare and ordain, That the ordinance adopted by the people of this State in convention on the twenty-fifth day of June, in the year of our Lord one thousand seven hundred and eighty-eight, whereby the Constitution of the United States of America was ratified, and all acts of the General Assembly of this State ratifying and adopting amendments to said Constitution, are hereby repealed and abrogated; that the union between the State of Virginia and the other States under the Constitution aforesaid is hereby dissolved, and that the State of Virginia is in the full possession and exercise of all the rights of sovereignty which belong and appertain to a free and independent State.

And they do further declare, That said Constitution of the United States of America is no longer binding on any of the citizens of this State.

This ordinance shall take effect and be an act of this day, when ratified by a majority of the voter of the people of this State cast at a poll to be taken thereon on the fourth Thursday in May next, in pursuance of a schedule hereafter to be enacted.

Adopted by the convention of Virginia April 17,1861.