I have a similar problem
Contact form would not send a message if user was logged out
error message was in a red framed box "Failed to send your message. Please try later or contact the administrator by another method."
I then logged in, could send a message successfully. After this action I could then send messages from logged out state and everything functioning OK again
I now modified this message in ..\contact-form-7\includes\functions.php giving an email address in lines 36 and 46