I frequently run out of storage on my iphone even after I delete photos, videos and apps. I go in my settings to manage my storage, but it doesn't affect anything. I have not upgraded my software because I don't have the storage left to do so, so I am running on 7.0.4 still.
Hmm. Lets get to the bottom of this. How many gigabytes does your device have available? Have you tried to wipe settings? Settings > general > reset all settings (be sure not to choose all content and settings).
Also do you currently sync with iTunes or iCloud? Test out after the rest and let us know how it goes.