wrong apple id

Use the Support article below to reset the new phone.

Selling/Giving Away Device.      

Then set the phone up using the desired ID. If you have a backup of the old phone, you can restore it. When you sign into Settings/Your Name/iCloud and turn syncing on, you should get your iCloud data back.

Backup - Restore your device from an iCloud or iTunes backup.    

Use the other Apple ID. If you forgot it, then see:  If you forgot your Apple ID  https://support.apple.com/en-us/HT201354.

You say: "Is the only way to change it on the phone to reset the whole thing and start all over again?" That won't work if you don't have an Apple ID that works. Don't try because if you do so you'll end up with your phone in activation lock. In this regard, did you or someone else  erase your device without signing out of iCloud first?