Where did you purchase the music? If you purchased the music In Google Play Music (or in the Google Play Store), you should be able to see it when you log into Google Play Music on other devices, such as your new phone or computer. If you log into Google Play Music on a desktop browser and don't see your purchased music (if you click the three dots on the track and see the option to Buy, then it's not purchased), you purchased those songs on another site/store. If you purchased these from another store that downloaded the MP3s onto your old phone, the MP3s would show up in Play Music as the player scans your device and pulls in all supported music files (even ringtones, unfortunately).
If that is the case, you can offload the songs from your old phone to a computer and upload them to your free Google Play Music library using either the Google Play Music Manager app for Windows or the Google Play Music extension in Chrome. Keep in mind that whatever computer you use to perform this action will take up a device authorization (any computer you use to download/upload music counts against your device limit).
Hope that helps.