[SEOvB]

If you were on a dedicated machine before you should have ssh access where you can perform the database back up, and move yourself even if the host only allowed 2GB downloads for whatever reason.

using a command like

mysqldump -u mysql_user -p -h localhost > databaseback.sql

scp databaseback.sql user@new.server.ip:/home/location/to/store on the old machine

If they still dont want to help you out since you're moving. Try to access your old site via ip.add.res.s and move your attachments from the database into the file system using the tools in admincp.

Then get a new database dump, and move the attachment folder to your new host.