My iPhone 4 will not receive messages, what do I do?

Just the other day my iPhone 4 stopped showing that I was receiving messages. I tapped on my "Messages" icon and the message that they have sent would be there, but it was like I opened it when I hadn't yet. Then it got worse… I would tap on the message that my phone thought I had read but hadn't, and most of our conversation from the past couple days would just be gone. I can still send messages out though and they are received.
I'm up to date on the latest software, but Every couple of minutes I get a message that pops up and says "Storage Almost Full… You can manage your storage in Settings," so I did. I have full memory in iCloud and I have 7.3G and 6.4GB used. All of my Apps are deleted and I don't have that many songs and pictures on my phone. As for messages, I have them dated back to 2011 just because I never delete them. So I decided I would finally delete them and they kept popping back up when I deleted them. So weird.
I backed up my phone, and rebooted it through iTunes, and what do you know? It still is doing the same thing. Thank you for getting this far and if you have any ideas to what I should do it would be so appreciated.

Do you think that someone else might be using your phone? Cuz msgs don't get naturally read. Do you have a passcode?

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