If you are like myself, I was actually very excited about the new iOS7 update. I have a newer 3rd Generation iPad, that I knew should run the new software fine, but I was worried about my older iPhone 4, but after using the new iOS7 on my iPad with no significant problems, and a little research I decided to take the leap, and downloaded the new Apple Operating System.
Everything seemed to go off without a hitch. I checked apps, you name it, and everything on my iPhone seemed to be working fine, for a few hours I noticed no major problems after the upgrade.
Then low, and behold, I decided to text a friend of mine who is in New York for a Indian Larry Block Party, and as soon as I started trying to text her, my texting didn't even show up. I couldn't even figure out what was going on at first, but after writing a word, or two, and waiting a good five, to seven seconds, my text showed up.
I was horrified at first because this was a MAJOR issue, one I knew, I would not be able to live with. I immediately began looking into going back to iOS6, and downgrading, and hoping my backup was sufficient for the process. I did a quick Google search to find others were having this exact same issue, and some of the post were made a couple days ago. I felt like an idiot for not doing a little more in-depth research before clicking the install update feature.
With all that being said I knew my phone was all backed up, so I decided to start doing some general reset's, and see if that changed anything. I first powered off my phone, and did a cold reboot, nothing, still same problem, but I decided to go into setting, and do a full feature reset, which will reset all wallpaper, volume settings back to the factory default.
Low, and behold this worked perfectly, so I decided to do the following video showing exactly what I did to get my texting working properly again, and at normal speed for my Verizon 32gig iPhone 4, updated to iOS7.
I hope this works for you as well.
Try Settings; General; Reset all settings.