Four prerequisites for domain name transfers:
1) WHOIS Privacy MUST be disabled.
2) Domain must be unlocked.
3) The contact email for the domain owner + technical contact needs to be set to something you regularly check.
4) You need to obtain the authorization code.
It is crucial that you complete all four steps - if you don't, your transfer will get hung up somewhere. If you haven't gotten any emails offering you a link to confirm the transfer you're trying to initiate, there's a hangup - most likely because you didn't complete one of the above steps. Ensure that you do, and resubmit the transfer. You can ignore the "Next Resubmission Date" ... just start the transfer over yourself.