Will music teachers be sinful if they remain in their jobs, having changed their teaching methods to permissible activities, after their request to change their job title was refused? And what is the ruling on their salary in this case?

Music is unlawful, based on the Prophet's (peace be upon him) saying: "There will be people from my Ummah who will seek to make lawful illicit sexual relations, silk, intoxicants, and musical instruments." Working with it and teaching it are unlawful, and money earned from it is unlawful. The questioner must try to transition to an administrative role or another activity, and pray and seek Allah's help in that. As for the salary he receives for not teaching music and benefiting students, we hope there is no harm in it, because his remaining is more beneficial for the students than his departure.