I was in a similar situation where I had an account in regular email@example.com and bought an account in Google Apps to have a new account in the new domain. I successfully moved Ownership from my old account to the new one.
First of all I gave Editor permissions to all my files to the my new account "firstname.lastname@example.org". Then I installed Google Drive for Mac (desktop application) and signed in with my old account. I waited until all my files synchronized then signed out from the application (Preferences... -> Disconnect account ...).
After this I had a folder called Google Drive in my home directory with all my files from Google Drive.
Then I created a new empty folder and called it New Google Drive. I opened the Google Drive application, clicked "Sign in" and logged in with my new account email@example.com, on the second step while I was signing in and changed the folder where the application should store synchronized files to the folder New Google Drive.
After the Google Drive application was ready, I copied all files (by COMMAND+C ) from Google Drive folder and put them (with COMMAND+V) in the New Google Drive folder.