How can I better convey the idea of "message sent"?

Judging by the context of the title bar, this seems to be a modal-type interface. When the user hits "send", you should wait until confirmation that the message has been sent. If it was, close the modal (and hide the keyboard) and add a small alert that it was successfully sent. The closing of the dialog signals to the user that something has happened, and the alert would signal what that was.

If it fails to send, don't close the modal. The user would lose their work, and would create a feeling of distrust in the app. Instead, give a small alert on top of the keyboard, signaling that the message failed to send.

If you're thinking only in terms of icons, then I guess you could use a paper plane for the "send" icon; this way, you can reserve the "check" icon for representing the "Sent" status.

I like the way WhatsApp resolves this:

1- sent 2- received 3- read

I think it's in the user's mental model when the color changes then it means "READ" so I would do it the other way around: from light grey to dark grey, or better change the color to green or blue to be even more explicit and you make sure something happened there. I think it's subtle the way you are doing it. It's just a personal opinion...

Usability testing is a great way to see which option is the best..

Once user hit send(Checkmark in your design), you should move to a sent messages screen or a conversation view. Also, hide keyboard.