I purchased more space but my device is still saying that it is full. What could be the problem?

Do you mean that you bought more iCloud storage or that you bought a new phone with more storage than your previous one and restored it from a backup? Because if the answer is iCloud storage, you have to make sure you have turned on the iCloud storage settings on https://icloud.com or in iCloud settings on your phone. If the problem is that you bought a new phone and restored it from a backup, there may be iCloud files such as photos that are automatically downloading to your phone. To check if this is the case, check what is taking up your storage in Settings > General > iPhone Storage.