The first sentence is wrong because "apart" describes the pair of you, not each individual. For example, John and Mary can be apart, but John can't be apart, and Mary can't be apart. Only the couple can be apart. Cross out "from each other" and the sentence is correct. "Away" works differently. John can be away, and Mary can be away, so it is correct to say they are "away from each other".

For the first sentence I would suggest also saying “ be apart “ instead of stay apart. I understand both sentences you wrote and they express the same sentiment. Great job!