The only way I can send a new text to someone and not to have the old text messages go along is to delete the old messages before sending a new message, which I don't want to do
You shouldn't have to delete the old text before sending a new one. Are you saying when you send a new text, an old text gets sent again with it? If so, try rebooting your phone.
Hold the top power/sleep button and the home button at the same time until the Apple logo appears, then release.