Should I restore my iPhone 5s?

There's a HUGE storage problem on my phone. I bought some tv episodes, but deleted them because they wouldn't all fit on my phone. The episodes deleted, BUT my usage said "12.9 GB used" and "0 bytes left" despite me deleting the videos. It even says I have no video content on my phone. I also have no pictures or music, and my most used up app is only 240MB. All of my apps are free, so should I restore my iPhone to get my data back? I feel it's the only way I can. I don't think it'll hurt my phone.

Yes. BUT, YOU HAVE TO START IT ON THE ICLOUD AS A NEW IPHONE! If not, if will still be corrupted (which is what you're dealing with) Be sure to do EVERYTHING though, Itunes, or this will continue to happen.

Yup.
Back up to icloud
restore through itunes
and restore from icloud back up.
you should be good to go from there.