Are you able to receive messages ? Also did you try resetting network settings ?
If it still not working, then you have to restore a fresh software on your iPhone from iTunes on your computer. Follow the steps below;
First download and install latest iTunes on your computer from here : https://www.apple.com/itunes/download/
Then connect the iPhone to your computer with the USB Cable.
Open iTunes application on your computer.
iTunes will recognize your iPhone. Select your iPhone in the top left (select iPhone icon)
Then in Summary section, click Backup and make sure "To this computer" is selected.
Then click Restore button and let it restore. It will take some time to restore the iPhone .
After restoring, you will get option to restore your backup. Select Restore from iTunes backup and all your files will be restored. Try to send messages now.
Please let me know if that didn't work, so we can continue troubleshooting.