Let's break down why 'I go without a bag to school' is correct and should be accepted, besides being an obvious either/or situation in English.
저는 - I
가방 없이 - without a bag
하교에 - to school
가요 - go
The word order -demands- that 'without a bag' comes before 'to school.'
There should be no penalty for writing it that way.
It can get pretty long as more parts are tacked on. And some expressions using it can be condensed into one word in English.
그 정당의 대표는 아무 반대 없이 재선되었다.
The party’s representative was reëlected without any opposition.
The party’s representative was reëlected unopposed.
그는 내게 말도 없이 급히 지나갔다.
He hastily passed by without even saying anything to me.
단어 hour는 첫 음이 대기음 없이 발음된다.
The word hour is pronounced without aspiration of the initial sound.
- 첫 음은 대기음이 없다. ~ The first sound does not have aspiration.
이 고기는 더할 나위 없이 연하다.
This meat is extremely tender.
- 더할 나위가 없다. ~ There is no need to do more (because it is already saturated or at its peak).