The main difference is that こそ is used to single out something as a primary "example" of something. It is usually only used to emphasize something "positive". It most often will replace は to add the emphasis. It's not a direct translation, but it might help to think of it in term of something like "especially" or "particularly".
- 音楽こそ命だ → Music is life
- これこそ待っていたものです → This is what I've been waiting for (more than anything else)!
- 今こそチャンスなんだ！ → Now (more than any other time) is our chance!
- 人間関係にこそ人生の価値がある → It's (particularly/especially) in relationships that one's life has value!
さえ singles out an "extreme", implying that the statement obviously applies to other examples. It most often carries a negative connotation, but not always. It is usually replacing も or を in the sentence, and can be thought of as "even". Actually, I believe every occurrence of さえ can be replaced by も, but not necessarily the other way around. Here are some of my examples from another さえ post that you can refer to for more info:
- 先週の病気は本当に辛【つら】かった。水さえ飲めなかった。 → "My illness last week was really bad. I couldn't even drink water!" → When one is sick, you expect them not to be able to drink alcohol, soda, etc. But water should be OK for most people. Even water, the most basic thing was intolerable for this person.
- えりこは親友の花子にさえ知らせずに外国へ旅立った。 → "Eriko left on a trip to a foreign country without telling even her best friend Hanako." → You'd expect her to not tell her co-workers, acquaintances, etc. But best friends usually tell each other everything, and she didn't tell even her.
- 山の上には夏でさえ雪が残っている。 → "Even if it's summer, there is snow left on the top of the mountain." → You expect snow to be on top of a mountain in winter, and some leftover in the spring. But usually by summer it's all gone, but for this mountain the snow remains even then.
So for your sentence, さえ is the correct choice. It sounds ungrammatical to me that have こそ in there. But regardless, さえ is needed for the overall meaning.
- 忙しくて昼ごはんを食べる時間さえない → I'm so busy, I don't even have time to eat lunch → If you're busy, of course you don't have time to go shopping, to the movies, etc. But this person is so busy, there's not even enough time for them to eat. Of course, this may be an exaggeration, but it still emphasizes just how busy they are.
- それが子供にさえ分かる常識です → That's such common sense that even children know it.