Are you hitting the menu button from within a text? What does it say in the blue bar right below your notification bar? When I am within a text message, and, it says Text below the bar, I cannot compose a new text using the menu button. However, if I hit back and it says Inbox, then when I hit the menu button, I get Compose.
Also, you can set Google Voice to be your default for texting, and, new text messages created will be through Google Voice. You can tell it's working because it'll have the Google Voice icon to the left of Compose in your text message when you go to send one.
I created Shortcuts to all the people I text and dumped them into a folder for easy texting. You just hold your desktop, select Shortcuts, then Direct Message. Much quicker that way.