Hi frogtree,
Thank you for participating in the Apple Support Communities. We understand you'd like to know if your daughter will lose her library and playlists if she switches from the Verizon Apple Music plan. As long as she uses the same Apple ID and she switches immediately to the other plan, she won't lose her library or playlists. She should be able to sign up before you cancel the Verizon plan. This article explains more about this: Connect Apple subscriptions from your wireless carrier to your Apple ID.
"If you already have a subscription from Apple
If you're subscribed to Apple Music, Apple One, or another Apple subscription through Apple and your wireless carrier, cancel one of your subscriptions. You can cancel your subscription from Apple. Or you can cancel your subscription through your wireless carrier."
Also, you can read more about current Apple promotions here to make sure she qualifies for the promotion: Offers - Apple.
We hope this helps.
Take care.