주세요? 드릴까요?

Hoya says 봉투에 담아 드릴까요?

Sori says 봉투에 담아 주세요.

드릴까요? is the clerk's offer ('shall I bag it for you?'). When you are the one asking, you say 주세요.

Learn it in Life in Korea · Unit 2

Hoya says 시럽 빼 줄게요.

Sori says 시럽 빼 주세요.

줄게요 is an offer — 'I'll leave it out for you'. When you are the one asking, the verb is 주세요.

Learn it in Life in Korea · Unit 3

Hoya says 카드를 충전해요 주세요.

Sori says 카드를 충전해 주세요.

주세요 attaches to the 해요 form without 요 (충전해 + 주세요). 충전해요 주세요 has 요 left in the middle.

Learn it in Life in Korea · Unit 1

Hoya says 아이스 아메리카노 한 잔이 주세요.

Sori says 아이스 아메리카노 한 잔 주세요.

주세요 gives the drink to you, so the drink is the object — it takes 을/를 or nothing, never the subject marker 이/가. In an order, drop the particle.

Learn it in Life in Korea · Unit 3

Hoya says 네, 필요하세요.

Sori says 네, 필요해요.

-(으)세요 honours the person spoken about. The clerk says it to you; you never use it about yourself. (Hoya says 필요하세요 all morning at work — that is why it slipped out.)

Learn it in Life in Korea · Unit 2

Hoya says 네, 목이 아프세요.

Sori says 네, 목이 아파요.

The -세요 in the question honours you, the person with the sore throat. In your answer it is still you, and nobody honours themselves: 아파요.

Learn it in Life in Korea · Unit 6