The best way to do this effectively I think is to backup all the databases and transfer via Site to Site VPN (setup between your location and East Coast Azure Datacenter).
To backup all databases, setup a SQL agent job with below step. Remember it's a CmdExec type and not a T-SQL statement.
sqlcmd -E -S $(ESCAPE_SQUOTE(SRVR)) -d master -Q "EXECUTE [dbo].[DatabaseBackup] @Databases = 'ALL_DATABASES', @Directory = N'C:\SQL Backups', @BackupType = 'FULL', @Verify = 'Y', @CleanupTime =312, @CheckSum = 'Y',@Compress = 'Y'" -b
And simple enough, copy all the .bak files from C:\SQL Backups to the remote Azure location. and use a restore script to restore all databases at once.
I recently found a tool (Teracopy) that does the copy/paste much faster than traditional Windows server copy functionality.
Hope this helps.