When you are trying to change the ID, are you signing out of your ID, and then adding their ID? Try signing out of your ID, restart the device, and then see if you can add their ID.
When changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion.
How to switch Apple IDs on your iPhone or iPad
How to transfer all files from one iCloud account to another
Apple ID email address or password - What to do after you change