The wealthy class in America, including that wealthy class of large landowners and farmers, did just fine during the Depression. Some economists suggest they did a little better than before, actually, as the money deflated and was worth more.
Many small family farms went broke and sold either to the bank or to the wealthy, who simply added to their wealth then when times later improved. The Dust Bowl and the end of the wheat boom of the 1920s created a demand for more wheat from the Midwest and West, and farmers in small communities were more insulated from the Depression because their economies were not that large to begin with.
You also found a large number of successful plantations (though they weren't usually called that anymore) in the South, where sharecropping was still the order of the day on large cotton and tobacco farms.
Clearly there were successful farms during the Great Depression. If there had not been any successful farms, there would have been no food for anyone to eat. It is certainly true that the farm economy was hit pretty hard by the Depression. This is especially true in the Dust Bowl area of the country. But this does not mean that all farmers went broke by any means.
Within my own family, my grandmother's parents lived on a farm in Idaho throughout the Depression and had enough money to send her off to college during that time.
In terms of big farms, though, we know that big farms were doing fine in California. The "Okies" flocked there to try to get jobs on such farms.
Lanterns on the Levee is a memoir about a plantation owner, his principles and ideals, and his plantation that existed before and during the Great Depression. It is by William Alexander Percy. It was first published in 1941. (A memoir is a book that recounts personal knowledge of the author.) Lanterns on the Levee is a very highly regarded memoir.
An excerpt from the memoir is found in vol. II of Major Problems in the History of the American South by Paul D. Escott and David R. Goldfield (1990, 110-114). This excerpt is about the plantation during the Great Depression. It was a cotton plantation in Mississippi.
Perhaps your public library can borrow one or both of these books for you.
The link that I have provided below, gives short reviews of the book.
You asked about plantations; plantations are large farms that devote all of their resources to the production of one crop for market. Coffee plantation, cotton plantations, banana plantations, and so on.