Of course they did. And by mid-1861 they (along with 7 other states) held state conventions. In each, delegates were chosen, and the conventions resulted in their being signatories to formal secession documents. Here is South Carolina's formal secession ordinance:
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acts and parts of acts of the General Assembly of this State ratifying amendments of the said Constitution, are hereby repealed; and that the union now subsisting between South Carolina and other States, under the name of the "United States of America," is hereby dissolved.
In each, delegates were chosen, and the conventions resulted in their being signatories to formal secession documents.
which, of course, were not recognized by the entity to which they were constitutional signatories in the first place, thus, they were not a ‘distinct nation’except in their own delusions, regardless of articles of secession not worth the paper they were written on...