Just disable the Privacy and unlock the domain before you do the transfer. See this tutorial:

http://hover.screenstepslive.com/s/hover/m/927/l/7788-GoDadd...

The issue is that exposes the OP's identity in Whois for a bit of time during the transfer.

Can you transfer to another internal GoDaddy account in a different name (that you create) and then transfer them out?

It would have to be to someone that you know and don't mind being exposed to said domain as in accordance with ICANN you can't use fake whois information.

Maybe put in fake whois information for the day that the transfer is happening. Once it's moved, re-enable the privacy. The info doesn't even need to be fully fake, you can use a throwaway email address so that you're still reachable.

Any change you make to your contact details will lock your domain for another 60 days. That means, you will be stuck for another 60 days before you can transfer the name away from Godaddy. http://help.godaddy.com/article/418 - Browse to the bottom and you can see it.

You aren't allowed to use fake whois information in accordance with ICANN.

I'm in the same boat same as you for one of my domains. I'd love to find a way around!