In general, accents should match the character, not necessarily the setting. If you make the setting clear, and give the character an accent that matches the setting, it will suggest the character grew up in that area and maybe has never moved away or traveled much.
On the other hand, for any character whose accent does not match the setting, it tells us they are from somewhere else. Or they might not have an accent from another place, but a particular way of talking that tells us about who they are. For instance, some Americans might say "I got no reason to lie", and others might say, "I am at a loss to conceive of why I would choose to prevaricate in this instance", even if they are from the same place - just depending on their upbringing and education, perhaps.
For your specific example, I was born in America and have lived here all my life, and I have no idea if people spoke differenlyt before WWI or how they spoke differently. I suggest you write with a modern American English style, and make the manner of speaking match the characters as they would talk today, at least in your first drafts.
The more concerned you are with accuracy, especially as a non-native speaker, the more you'll want to get help from someone who knows. This might be a fellow writer who has a better understanding of American English than you, or a historian who knows how people used to talk, or a linguist or anyone who has specialized knowledge.
If you want to do it all yourself, then I suggest you read books that were written in the time period your story is set in. One thing I have noticed in early 20th century American literature is not so much people talking differently (although the words for many things are different), but the morality seems very different from today. Reading both fiction and non-fiction from the time period in question will give you a lot of information about how things were.