I have not biked in the south and I am going from San Diego to Pheniox. Do I need special tires? Right now I have your basic tire for a touring bike Trek 520.
I would suggest that a bit of extra care with where you wheel your bike when off pavement goes a long way toward preventing flats from goat head thorns. They are likely to be just about anywhere you wheel the bike off of the pavement for a rest or other break. Try to be careful where you go and check for thorns when starting out again.
For tires I like Continental Ultra Gator Skins and beyond the care mentioned above I make no special allowances for riding in the southwest.
Having grown up and lived my whole life in the east, when I rode out west I got a lot of flats before I figured out what goat head thorns were and where you find them.