Hello Tabatha,
Thank you for using Apple Support Communities! I understand that songs in playlists on your iPhone are playing on shuffle. I would be happy to help provide some steps to take so you can play your music as expected again.
Since you've already checked that shuffle was not on for your playlists, let's also confirm that shuffle has not been enabled on the Now Playing screen:
Play, repeat, and shuffle music - Apple Support
Specifically, follow the steps in this section to ensure that Shuffle is turned off:
When playing music on your iOS device or Android phone, a player appears at the bottom or lower-right corner of the screen. Tap the player to open Now Playing. From the Now Playing screen, you can control playback or volume, turn on shuffle or repeat, download the song, and more.

Open the Now Playing screen and scroll down.
Tap
to set Music to always shuffle a list of songs selected. Any album, playlist, or set of songs will play in a random order.
If shuffle is on, you'll see
. To turn it off, tap the button until you see
.
If Shuffle was on from the Now Playing screen, let us know if the behavior persists after turning it off.
Cheers!