One thing I vaguely remember from a client I was working with (who had a domain on Godaddy I was trying to transfer) was that they had accidentally signed up for some sort of GoDaddy Extra Domain Security. This was blocking all attempts to transfer the domain.
The client had to call GoDaddy to get that cancelled before we could proceed.
Also I seem to remember GoDaddy has a little section in the Domain Management tool that shows if a outgoing transfer has been requested for a domain.
You might want to wait a day before doing anything…check all emails again…check that section in GoDaddy…if still nothing then try the transfer again.
If you paid once, it should just be a credit in your DreamHost account. If so, it will let you use that “credit” next time you try (rather than having to pay again)