I think you are right - the Confederates would have used the same judicial system that they were accustomed to when in the Union. After all, there would have been no need to reinvent the wheel. I am curious to what extent a judicial system was in place - did they have a judge advocate, an appeals process, etc.
Also, one civilian court-martial I have looked at occurred in Memphis, Tn. The accused, Kate, was living in a brothel (or a house of "ill-fame"

. She was tried for disobeying a lawful order by living in a brothel (I am not certain what the order was), found guilty, and sentenced to be exiled to Illinois and her property was confiscated by the government.