Both 10월 달에 시험 있어 and 10월에 시험 있어 are commonly used. And same goes for 8월 27일 날까지 and 8월 27일까지. Adding 달 and 날 like this looks redundant and illogical, but it is done all the time. I think the reason is in part because they make things clearer. In Korean, the pronunciation of 달 and 날 stand out more than 일, 월, or 에. So adding them makes it sound stronger. Adding these redundant words are probably more prevalent in speeches and informal writings, and the official recommendation seems to be not adding them, But I think new learners can use both types of expressions freely without worries.

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