The US Constitution (article 4, section 4) says "The United States shall guarantee to every State in this Union a Republican Form of Government", so they had to have some popular elections. But maybe I misunderstand the question?

Under the South Carolina Constitution of 1790, the people did elect the lower and upper houses of the state legislature, and by the U. S. Constitution the U. S. representatives ... but not a whole lot more. The legislature elected a figurehead governor, who had no veto, a two-year term, and could not serve again until after four years. The legislature elected the U. S. Senators, but that was formally true of all states until 1913 (though by then 29 states had state referenda to direct the vote of the state legislature). The legislature elected the "judges of the superior courts, commissioners of the treasury, secretary of the State, and surveyor-general", and other local officials like tax collectors[1], but I haven't found a list of who was being elected by the voters.

And yeah, the legislature elected the presidential electors. The U. S. Constitution (article 2, section 1) just says "Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors ...". In 1789, the voters elected electors (more or less) in 6 states. By 1832, only South Carolina retained election of electors by the legislature[2]. At the Confederate constitutional convention of February 1861, South Carolina's representatives tried and failed to get in a requirement that presidential electors must be chosen by the legislature[3].

[1] Greenberg, Kenneth S. “Representation and the Isolation of South Carolina, 1776-1860.” The Journal of American History, vol. 64, no. 3, 1977, pp. 723–743. JSTOR, JSTOR, www.jstor.org/stable/1887238.

[2] Dixon, Robert G. “Electoral College Procedure.” The Western Political Quarterly, vol. 3, no. 2, 1950, pp. 214–224. JSTOR, JSTOR, www.jstor.org/stable/443484.

[3] C. Blease Graham, "South Carolina’s Constitutions", https://web.archive.org/web/20080225080851/http://www.cas.sc.edu/poli/courses/scgov/Articles/SC_Constitution_History.htm .