Question about transferring domains

Well, after my site went down again last night due to my crappy host, I’ve decided it’s finally time for me to make the move to DH I’ve been meaning to do.

One thing I’m concerned about, however, is transferring my domain. Reading the Knowledge Base article on it, I obviously need to contact my current host to make this happen.

But one thing I’m wondering is, can I still upload files to my DH account here before actually having my domain transferred? I ask this because I’ve got a MySQL database that I need to copy over to DH’s servers before my site will be operational. I wouldn’t want people to visit my site via its domain, and see, you know, nothing. Would there ever be a point at which I can access my current host’s files and copy them over to DH’s hosting? In other words, would I be able to access both accounts at the same time, until I told my current host to switch modify their DNS info?

Really, I’m trying to make the transfer with the least amount of downtime as possible. Can anyone provide me with a little more insight into how the entire transfer process takes place?