I have two iPhones using the same Apple ID. I have tried to log one in under a different ID but it changes the ID on both devices

It sounds like maybe you are using the instructions for changing the email address used for an account instead of signing into a different Apple ID.

Do this on one phone:

Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac --> Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac - Apple Support

Then this:

Click on this link to learn how to 'Sign in with your Apple ID' --> Sign in with your Apple ID - Apple Support

1. Open the Settings app.

2. Tap Sign in to your [device].

3. Enter your Apple ID and password.

4. If prompted, enter the six-digit verification code sent to your trusted device or phone number and complete sign in.

If another Apple ID is signed into the on on which you are trying to sign in you may need to enter the password for that ID to sign out of that ID first.