I am paying for more storage but my new phone says it’s run out of space

Go to settings and ICloud and make sure photos are being uploaded to iCloud and are not being stored locally on iPhone.

This article is very helpful:

About storage on your device and in iCloud

https://support.apple.com/en-us/HT206504

This support article explains the difference and also has the “Fix”

Joy

Icloud storage is not Iphone physical storage You are not increasing the device storage with Icloud it an mirroring data option to server from device so Your device has 64 gb physical storage and if You have 2TB icloud storage plan it not means that You have 64gb+ 2TB storage.