Hey @chase.freeman . I had the same issue and have been discussing with @Jaime as there might be something wrong with the gateway. In any case, the why I solved it was stopping the app (via settings > force stop) and removing the send SMS permission. Open it again and configure with URL, username and password (make sure the user has a phone number in its specific section of the user config and it matches the SMS sender) then try again and you should see things happening in the logs.
In any case, we will review the app here: Log in - DHIS 2 JIRA