Unfortunately, that day overlaps with the day I promised to do something with someone else.
That's too bad. I already made plans to see someone else on that day.
この場合、次のような言い方ができますよ。
ーUnfortunately, that day overlaps with the day I promised to do something with someone else.
「生憎、その日は他の人との約束が重なっているんだ」
that day overlaps with the day ... で「その日は…する日と重なる」
overlap を使うと自然で良いですよ。
ですが、次のように言う方がもっと自然です。
ーThat's too bad. I already made plans to see someone else on that day.
「あー残念。その日はもう他の人と約束しちゃったんだ」
ご参考まで!