Hi Lori-GracesMom,
Thanks for posting in Apple Support Communities.
You mentioned that you share Apple Music using Family Sharing, but it sounds like you're sharing an Apple ID if you can see each other's playlists. You should have different Apple IDs when you go to Settings >[your name] on an iOS and System Settings (Preferences) and look at the very top of this panel.
You can create a new Apple ID, if necessary.
Create your Apple ID on iPhone, iPad, or iPod touch
You can create a new Apple ID when you set up your device, or set it up later in the App Store.
Create your Apple ID when you set up your device
1. Tap "Forgot password or don't have an Apple ID?"
2. Tap Create a Free Apple ID.
3. 
4. Select your birthday and enter your name. Tap Next.
5. Tap "Use your current email address," or tap "Get a free iCloud email address."
6. Follow the onscreen steps to verify your email address, create a strong password, set your device region, and set up two-factor authentication. If you choose to skip this step, you'll be prompted to do this later in Settings.
How to create a new Apple ID - Apple Support
If you're not sharing an Apple ID, are you choosing to do a one-time share?
Provide more details on this issue.
Regards.