I do not know if it was an unwillingness to change or an inability to change, but you can certainly argue that the South's relative lack of development made the Depression worse there than in other parts of the country.
Even after the Civil War, the South continued to be a much more agrarian and unindustrialized area. Because of this, the area was very susceptible to changes in demand from the places that bought the things the South produced. Since the South was still heavily dependent on a few products, a drop in demand for those products could devastate its economy.
If the South had had a more diversified economic system, the Depression might not have had such a severe impact in that region.
Besides having its limited industry and transportation systems destroyed during the Civil War, the South also faced a general lack of interest on the part of northern citizens and government. Little to no investment was made in economically reconstructing the South, which weighed them down for generations.
The South also bears some responsibility for this. When they had the chance to re-engineer a new, more diverse economy after the war, they instead reverted to the same landowning/virtual slave system of agriculture run by white plantation owners and their sharecropper/tenant farmer workers. This left little money in the hands of the general population, and kept the economy from growing beyond those boundaries.
There were several factors that had contributed to the south being hit extra hard during the great depression. The Civil War left the south economically deprived of a means to its previous form of funding. The cotton industry could no longer rely on save labor. In addition, the mills had beendestroyed by the military maneuvers.The south being an agricultural community took a long time to rebuild the land to a level that could produce an income again.
Another problem was the situation with displaced black people. They were still struggling to find a way to eek out a living among the competition of white people in a very prejudiced economy. The few jobs available went to white people.
The north had factories and other means of income than the south. The south had stayed to its roots and the goods that the provided were not purchased or produced as readily on the market.
One benefit that the south did have though was that many people during the depression had at least a food supply of vegetables that were grown in small to mid sized gardens. I recall great grandmothers telling me about northerners coming down looking for jobs. They couldn't give them work but they could feed them all the turnips and potatoes they could eat.