Hi,
Sorry to hear you are having issues with your device storage.
iCloud+ Storage does not back up Music in terms of saving a copy of it to the cloud. This applies to purchased content, imported content, and shared purchases. Purchasing an iCloud+ storage plan would be beneficial to have for other content and data, but not for this content type.
iCloud Music Library/‘Sync Library’ does keep your Music ‘backed up’ by associating it to your Apple ID and storing it in iCloud if you use Apple Music or iTunes Match. This means if you sign into any device with your Apple ID, all of your Music will be available for download to that device without having to actually sync the device via iTunes on a Mac or PC.
That being said, having these services will not help your device storage issue.
I would recommend managing the content you are syncing to your device or purchasing another device with a higher storage capacity in order for you to easily access all of your Music without running into these issues.
Hope this helps!