IPhone :: 4s Won't Save Photos

My iPhone 4S won't save photos taken with it, or sent to it. The phone is only a little more than a week old, and as far as I can tell, has never been able to save a photo, I only just noticed the other day because I was busy learning the other bells and whistles.

I can use the camera feature to take a photo and immediately send it as a text, but I cannot save any photos taken. It pretends like it saves them, and then they disappear.

I've googled around, and most places lay out a plan that involves creating a back-up, doing a hard reset, and then restoring the phone. However, each of these make a point that when I'm restoring the phone I should not "set up a new phone" and I should not "restore with the backup" I've created, but as far as I can tell, these are my only two options.

None of the articles about this that I've found have mentioned the iphone 4s directly, so maybe the steps have changed since they were originally written?

Info:
iPhone 4S