Yes. It will care in a "it will force them back into the ranks if it can" sort of way.
The CSA government that passed a draft act in the second year of the war is not the kind of government which will regard desertion as a reason to get realistic, its the kind that will regard desertion as a reason to get tough.
They had problem OTL with desertion. What kind of problems will they have when they find out that they could go home if the government wouldn't do staggeringly stupid things?