I'm having the EXACT same problem with my 4S. I had a ridiculous 2GB of messages. I restored the phone (after resetting with no effect), and although under my usage settings it says I only have 65MB of messages now, I'm still experiencing the same problem when I try to delete my texts. My texts are not set to sync with iCloud or anything.
Oddly enough, the very bottom text conversation actually does appear to delete after I reboot my phone. However, deleting text conversations one by one in between rebooting my phone doesn't sound like an ideal solution. I've had this problem ever since the iOS7 update. I was hoping the subsequent software updates would fix it, but they haven't at all. I love everything about iOS7, except obviously this. Please help!
Also, thanks for that troubleshooting link, but I don't think any of us are actually having problems sending or receiving texts, just merely deleting them.