Doing the above would erase the phone. If I understand correctly, your Apple ID works normally everywhere except when you try to sign into the phone. If that is correct, I would try contacting Apple ID Support. I have read a few posts where a person's ID was rejected only on one device, even though it worked everywhere else. Following the support links are some suggestions for when you can swap the ID.
Apple ID Support
Apple ID Support (2)
Apple Support by e-mail or chat Work through the screens to get a chat option.
Apple Support using Twitter
Or fill out and submit this form. The form is for iTunes, but it will get passed to the ID Support section.
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. Also set up Family Sharing to keep access to your apps/purchases.
How to transfer all files from one iCloud account to another
Apple ID email address or password - What to do after you change
Family Sharing
Family Sharing - Set up
