If you read some of the newspapers written at the time, you might notice that the level of literacy expected of the reader was higher than it is now. The level of discourse was higher.
If you want to represent some of the excesses of the period, F Scott Fitzgerald wrote quite a few fine works that capture the era. John Steinbeck wrote well of some of the miseries of the time, the hardships overcome and the obstacles faced.
My mother grew up during the dirty thirties and lived with poverty, drought and much love. My grandfather became a farmer to provide for his family, but knew little about the task he set himself. He worked with horses - not the type often shown as draught horses, but any horse that could fill that role. One neighbor was a bit more extreme in his methods than my grandfather. He found a badger in a slough on his land and decided it had to go. He and the neighbor dug after that badger for hours, finally catching and killing it. The next evening, they were invited to supper by the neighbor and enjoyed a fine stew. Meat of any kind was a treat, so it was eaten eagerly. Only later did they learn it was badger stew.
Read newspapers of the time and try to find ones for the locations you want to feature. Language was used more precisely as people were encouraged to speak well.
Do not assume the people of Oklahoma will talk like the people of Kansas. There are similarities, of course, but differences as well.
An American can, after a few minutes conversation, ask the other which part of Kansas they come from - some have such a fine ear for accent that they will know the town itself. No one ever asked what state they come from - that is clear from speech. Idiom and accent combine to make each state unique in speech and pattern, though the similarities will outweigh the differences. If there is an error or confusion, it is usually caused by the person in question having lived multiple places and his speech reflects the blended history of his residence.
People from that region tend to be polite and hospitable. Of course, some of the most vicious criminals of the time hailed from that area.
Choose one state and one era. It will simplify matters for you and eliminate the risk of some reader from Nebraska noting you confuse them with Oklahoma.
There was a tv show about a Depression Era family called the Waltons. My father was impressed by the level of detail that went into it - including costumes. Everything was clearly handsewn as sewing machines were less prevalent then.
I am sure you won’t do this, but please get the geography right. If you choose Kansas and write about characters going to the capital of Kansas - don’t call it Kansas City. Such an error could be jarring and the reader might continue only to catalogue your errors.
The 1904 Sears catalogue is an amazing resource for items that could be in any home. While it does not have dialogue, the item descriptions are subtly different from what one would expect now.