IPhone 5 storage completely full and can't figure out to fix the problem?

Question

I got the iPhone 5c in August of last year. Previously, I had owned a flip-phone and only a flip-phone, so you can imagine that it was a big change. Though I know there is so much more that it can do (and so many things I have yet to discover), I mostly appreciated the phone for its photo and Internet capabilities. Only a few weeks after I got the phone, I went to a concert and, as you can imagine, the videos took up a lot of space on my phone. I also have a bad problem with saving screenshots, so I quickly filled up space with pictures. Once I realized it was reaching an overwhelming amount (and space was getting limited), I backed-up my iPhone to iTunes on my computer and deleted around 800 pictures. It barely freed up any space. I did this once or twice until one time I tried backing it up again and it would not let me, saying that there was not enough space to do the back-up. I don't quite know how that works, but after that, I just ignored my problem. Well, then my space was dwindling down close to zero. I deleted most of my apps, though it did not free up much at all because 11.4 GB of my 13.2 GB phone is from Photos/Videos.
My phone reached 0 bytes available in, I'm guessing, November of last year? To my surprise, I could still use some of the features on my phone. Internet, screenshotting (though I could no longer take pictures), and some of the apps (though they often would not load fully). This didn't last for long. One day, I wasn't able to open hardly any of my apps. Then it transitioned to my Internet closing out of itself every now-and-then. Well, currently (and I've been having these problems consistently for the past month or so), I can not open any apps. If I try to go to a website, it closes out of itself almost every single time. I never know if my texts go through because it never finishes the sending process (sometimes people say they get them, sometimes they don't). I cannot receive texts. I can't even try to delete my photos/videos because when I click "delete", it goes to my Home Screen (again, it closes out of itself) and when I check, the pictures/videos are still there. For the past month or so, the only thing my phone can do is call. Which, you know, defeats the purpose of an iPhone. Basically, I have gotten myself into quite a conundrum.
Is there a solution to my problem that does not involve restoring it to factory settings?

Answer 1

Thank you for your reply! I just tried importing my pictures again. iTunes gave me the same message - that the storage was too full. I looked under summary, and I don't quite exactly know what usage numbers you are looking for, but it does say it is over capacity by 190.8 MB. However, when I plugged in my phone to the computer, it started to import pictures automatically into Dropbox. I just recently got Dropbox, so I did not have it the other times I tried to import. I managed to get the pictures onto my computer and, by some miracle, was able to delete a couple hundred from my phone. It also let me delete the long videos on my phone, which I've been trying to do for months. I went to my Storage and it now says I have 6.6 GB free (which was so amazing to see that I had to take a moment). I can take pictures now. I quickly updated a few apps and the space is still good. However, there are some apps that it still will not open (as in I open it up, then one second later it closes out). These seem to be the social media apps-- Twitter, Instagram, Facebook. Should I just delete them, then re-download? That will not mess anything up, correct? So far, Safari is working, which is a major relief.
How would I go about syncing things like contacts?