[Stu]

Do you have CPanel?

if you do you may be able to use a cron Job. I have CPanel on one of my shared accounts but I dont have Shell access on it, by using a Cron command I can backup and restore with no problem.

create a gzip backup of your DB then upload the file using FTP to the new server. to your Home Root.

Assuming your Database backup file is called DILBERT.gz
Re-Name DILBERT and the following

USERNAME to your Database Username
PASSWORD to your Database Password
DATABASE NAME to your database name

then run the following Cron Job

gunzip DILBERT.gz ; mysql -u USERNAME -pPASSWORD DATABASE NAME < DILBERT ; rm -f DILBERT

The above command will unzip the .gz file, restore the sql database and delete the gz file [b] NOTE: the database must exist, if it doesn't create an empty one first.

I've assumed that you have shell on your present account, to make the backup. If you dont then use cron with the following command to create backup

mysqldump --opt --user=USERNAME --password=PASSWORD DATABASE_NAME | gzip > /home/YOUR_DOMAIN/DILBERT.gz