I would expect the abolition of slavery in the CSA would be on a state by state basis, given the design of states rights over central/federal authority. Even in agriculture as it progresses from labor intensive with relatively basic machinery to more mechanized with more complex machinery slavery becomes less useful. Slave labor is rarely efficient, and in general slaves are not highly motivated to take special care of of more "delicate" machinery - powered farm machinery needs more care than horse or human powered equipment, and costs more to fix. The experience OTL showed that slaves employed in industry who became skilled workers were given more slack than plantation hands, simply because without deliberate sabotage they could gum up the works quite easily and their expertise and skills had special value. As slavery becomes uneconomic state by state, those no longer in servitude will face highly restrictive black codes that make South African apartheid look like a children's party. The problem will become how does the CSA deal with the fact that as long as slavery exists and slaves have economic value, producing more slaves is a "good thing" which means the large black population becomes an even larger percentage of the population.
As far as international pressure goes, I very much doubt you'll see the sort of pressure through boycotts, economic isolation etc that was used against South Africa OTL (and a century after the ACW btw). The major international "pressure" will be the fact that the major southern export, cotton, will face increasing competition from other sources. Other than cotton, the south had very limited exports, and as an independent country will have more cash outflow for imports than when it was part of an internal market. As oil comes online later on that will help the CSA balance of payments, but the CSA will still be an economy based on extgraction of natural resources which is not a good thing.