How can I change the organizer of our family sharing account?

You need to disband the existing family using the steps in the article below. The new organizer can then set up the new family and invite the family members to join the new family. If there are children under 13 years old, the new organizer needs to leave the existing family, set up the new family, and then the under 13 years old children can be transferred to the new family. Then disband the old family and join the new family.

Family Sharing -  Transfer a child to a different family group - Apple Support     

Family Sharing - Leave.  

You are welcome. I think it will still work, but you may have to cancel his subscription, and have you start a new subscription.

Apple Music Family subscription.   

You are welcome. If you have to change the subscription, I think so. I don't know very much about Apple Music.