Can I change my Apple ID without losing data?

Were you using the same email address as the ID for purchase and for iCloud? What I mean is that I use the same email address as my Apple ID and my iCloud ID. If you were doing the same thing, you can sign out of the old iCloud ID and sign into the new iCloud ID.

Take a look at this.

What to do after you change your Apple ID email address or password - Apple Support

Update your Apple ID and password on your iPhone, iPad, iPod touch, Mac, PC, and Apple TV.

iPhone, iPad, and iPod touch

Go to Settings > iCloud.

• If you're signed out, enter your updated Apple ID and password. The data in iCloud will automatically appear on your device after you sign in.
• If you're still signed in with your previous Apple ID:
  1. Scroll down and tap Sign Out, then tap Sign Out to confirm. If you're using iOS 7 or earlier, tap Delete Account, then tap Delete to confirm. You might need to enter your password. Your data will remain in iCloud and will be updated on your device when you sign in to iCloud again.
  2. Enter your updated Apple ID and password.

If you updated your Apple ID email address before signing out of iCloud, and your device repeatedly asks for the password to your previous Apple ID, you can temporarily change your Apple ID to sign in.

If you got a new email address and you used your old email address as your Apple ID and iCloud ID, you can change the ID and keep all content and data.

Change your Apple ID - Apple Support

When you change your Apple ID, you will not lose any data. If you create a new Apple ID, that will cause you to have to start over and lose everything that you purchased with that ID.