Memory Problems

I have an iPhone5. It says it has 0 bytes free. I have deleted a couple apps with all their files, and also un-syced photos in itunes, asking it to delete previously downloaded photos. The phone still stays it has 0 bytes free. The apps and thousands of photos are gone, but it hasn't appeared to make a difference to the available memory.

The phone is also experiencing additional bugs--tech support both at Verizon and at an Apple Store (one of the general Q&A people, not Genius Bar) said this could be caused by the lack of free memory, so I'm not sure if that is the cause or if both aspects could be caused by some other problem. My call log is not catching hardly anything. Even when a call (taken or missed) does appear in the call log, it doesn't stay there. An hour later it may be gone. Likewise some voicemails have deleted themselves. Other test voicemails never appeared at all. Apps sometimes quit immediately on opening, or a couple seconds into loading. This includes trying to backup to iCloud (it gave me an error message in one screen, when I finally made it through to the screen with "Backup Now" as soon as I hit it, I was shunted to the main desktop screen as the entire "Settings" closed on me).

I tried a soft reset which temporarily made things like receiving calls and voicemails and opening apps run more smoothly again. But the operational errors began again within an hour or so, and it never has budged on the "0 bytes free" issue.

(By the way--I'm not sure what iOS I'm running. I know 8.4 wants to install but, of course, there is no memory to do so. So, 8.3 is my best guess of what's currently on here).