insufficient memory - Getting "Storage space running out" message in Galaxy S4; *#9900# does not show any menu

I have a Samsung Galaxy S4. I am getting the "Storage space running out" error message in my notifications bar, and some applications have quit working or updating. None of the usual solutions I've read about elsewhere on the site have helped me.

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The most common solution to this problem is to access a hidden menu by dialing *#9900#. When I try this, however, the menu does not appear:

Another common solution is to root the phone and delete /data/logs. Looking at what's taking up so much space on my phone, it does seem likely to me that the problem is my phone has lots of accumulated log files that aren't being deleted. I would rather not root my phone, however. I am happy with the functionality my phone has, even with bloatware and tied to my provider, and I don't want to take the time to root my phone for this one problem if I don't have to. =P

Finally, a third option I've read is to delete apps that are taking up a lot of space. I've attached a few screenshots below showing how my phone is using up space, though, and I believe that they show that my real problem is that there's something my OS is doing wrong, perhaps creating a lot of log files and not deleting them to save space. (Besides that, I've already tried deleting some apps I don't use often. The overwhelming majority of my apps don't take up much space at all, and the biggest space-wasters are also the apps I use the most and the reasons I have a smartphone.)

Is there another way for me to get back more storage space without rooting my phone?

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If necessary, I'm willing to plug my phone into my computer and use developer tools to resolve this issue.

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Here are a couple images showing that the biggest gain I could get for storage would be to delete some system files (which is why I'm guessing that the problem is accumulated log files). I have over 6GB that my phone categorizes as "Miscellaneous files:"

And of the files my Android includes in that category, over 6GB are generic "System memory" (again, I'm guessing logs) that, unfortunately, I can't interact with or delete from this menu: