Hmmm...did they? My family were New York City copperheads and one wrote a letter trying to get out of the Union Army. New York City was practically a Confederate stronghold! And of course you know about the draft riots there.
There were even some Southerners that stayed loyal to the Union. One of the great Union generals, George Henry Thomas was a Virginian. But by and large, it was split between North and South. A lot of people in the South decided “Lincoln is not my President!” Where have we heard that before?