The Greeks and Romans didn't grow much cotton either. I guess that means they didn't have slaves?
Seriously, slaves could be applied to other crops than cotton. They could also be used in mining, lumbering, grazing, construction, shipping, and manufacturing.
The ancients used slaves for many of those purposes. I'm pretty sure that New World slaveowners, many of whom knew Latin and Greek, were aware of that.
P.S. Cotton was being grown in Arizona something like 5000 years ago. Not much, probably, but some. In the Old World, the Egyptians were using irrigation to grow cotton at around the same time.
I don't get it -- the mechanical cotton harvester was supposedly just around the corner in 1865, but irrigation technology was beyond the means of 19th century man?
The truth is that cotton growing wasn't necessary for slavery to exist and slaveowners knew that. But if they absolutely wanted or needed to grow cotton in dry areas where the temperatures were suitable for cotton production, they'd find a way.
I'm sure they could have, but if such usages were profitable, why didn't they? I don't know too many people who willingly leave money on the table. Most people try to maximize their investments.