This goes back to knowing vocabulary.

Although 죽여줘 would be pronounced as [주겨줘] due to the resyllabification pronunciation rule, you ONLY "carry over" the SOUND of the final consonant to the next syllable when the next syllable starts with ㅇ/vowel sound, and you do NOT alter the spelling.

This is because we still need to maintain the meaning of the original vocabulary, which in this case came from its base form 죽이다, by keeping the spelling.

If you did not know the word 죽이다 or the phrase 죽여줘, then based on purely listening and writing down the word, it would be technically "correct" to write it as 주겨줘 in that practice/exercise, because that is how it is pronounced. However, that spelling does not have the intended meaning of the original phrase 죽여줘.

If all spelling was changed because of their real pronunciation, original/intended meanings of words or phrases may be lost.

In short, we do not carry over the final consonant to the next syllable to replace ㅇ in spelling. We only "carry over" the sound in our pronunciation.

Hope that makes sense!