Hey there Itsssarahhh,
Taking a look at the steps found in, Use Shazam with Apple Music, Snapchat, and more indicate following the steps below should help get you up and running.
"Apple Music
Tap the Apple Music button near a Shazam to listen to that song in Apple Music. If you’re an Apple Music subscriber, songs that you've Shazamed can be automatically added to a playlist called My Shazam Tracks in the Music app. On iPhone or iPad, swipe up to My Music from the main Shazam screen, tap the Settings button
, then turn on “Sync Shazams to Apple Music.”
You can also add a Shazam to another Apple Music playlist without leaving the Shazam app. Just tap "Add To" next to a Shazam, then tap the playlist. If you’re not already a subscriber, learn how to join Apple Music.
If you can’t add songs to a playlist, make sure to sync your music library with Apple Music."
Have a good one!