They absolutely were. In fact the Confederate constitution was largely a copy of the US constitution, with only a few words and lines changed here and there.

On the national level elections were held for president in 1861 with Jefferson Davis winning a landslide victory (and all electoral votes) on a national unity ticket. Elections for congress were held in 1861 and 1863. While local elections more or less continued as usual throughout the war.

One can argue about exactly how free and fair those elections were, but they were roughly as free and fair as elections were pre-war. One can argue that the Confederacy wasn't democratic, but nearly all those arguments would work equally against the pre-war United States. All these elections were generally contested. And while there was a great deal of sentiment toward a war unity, there was plenty of robust debate. Congress regularly fought amongst itself and against Davis' administration. So did the individual states.

And if they were, wouldn't that break the rule that two democracies can't go to war with each other?

Almost certainly. That rule is either false, or has to be so narrowly tailored as to be meaningless. There's a good couple dozen exceptions to that "rule." For example you can argue that the Confederacy isn't a "proper" democracy because it had non voting groups like slaves and women... but then you're narrowing the definition tightly enough that neither the US nor the Confederacy counts as a democracy.

And that's how such arguments usually go. Someone will mention the Spanish American War. And the argument quickly devolves into an argument about Spanish politics and the power sharing agreement in place at the time, and whether the parties exercised proper electoral procedure etc. Or the arguments switches to Jim Crow and whether the US counts as a democracy when a large section of the populous was disenfranchised, or Tammany Hall and political corruption in Northern cities and States.

The problem is if you disqualify all democracies because they aren't "pure" enough, then the rule doesn't mean anything. If no one qualifies as a democracy, then it's meaningless than they've never gone to war with one another.