The documentation in the link stack247 posted is now outdated. To change the target subscription for a database, you have to change the target subscription of the server for the database. There are several ways to navigate to the server, but here is just one example:
- Click on the database tab
- Click on the name of the database you want to change
- In the dashboard view, click on the link to the server in the quick glance section. This is on the bottom right corner of the view.
- In the dashboard view, click on the Change Subscription link in the quick glance section. This is on the right side of the view.
- Change the Target Subscription, and click the check button.
As far as changing the subscription for a website or cloud service, I would do what is recommended above; and republish the website or cloud service. When you publish, change the settings to target a different subscription by clicking on the previous button.
---------Update----------
If you have a bunch of assets that need to be moved to another account, create a ticket with the Azure helpdesk. You will have to do the above and switch out the accounts for the databases/servers, but Microsoft is able to run a script on their end that moves everything. Depending on the size of your assets, it may take a little longer. Fortunately for me, it was only a day to move everything. I would recommend doing this instead of publishing everything a second time.
https://manage.windowsazure.com/?getsupport=true