The South's biggest advantage was that they were fighting for their homes and their way of life. Had they attempted conquest, they would have lost that strength. They would have had no chance of conquering the North.
Of course, they couldn’t. The decision to attack union forces which had withdrawn from 5 bases to Ft. Sumter was suicidally stupid. The big problem facing the South wasn’t invasion from the North, but what to do with millions of slaves as slavery became an increasingly obsolete economic strategy.
Lincoln’s plan to let slavery “wither on the vine” was much better for Dixie than the Confederacy’s plan to make the Southwest slave-dependent: The Great West wasn’t endless, and the region from the Ogallal Aquifer West could never have displaced southern cotton, or for that matter, rice, hogs or sugar. The plantation owners would have been much more wealthy controlling land as the US immigration population bulged and they gradually reduced their dependence on slavery.
But the problem was that the South feared retribution as slaves gained their freedom; they couldn’t imagine their slaves being their countrymen.