Are you moving to a dedicated server with the same host? If so, ask them to create a MySQL dump (assuming thats the db you're using) and transfer it to your new dedicated. Then, you can import your database into your MySQL server using:
mysql dbname -u User -ppassword < mysqldumpfile.sql
I had to leave the country for a few days and so I apologise for my non-response.
I hooked up with the guys at SupportMatrix and as a direct result we are now up and running on the new server. Communication was a breeze, once I had an internet connection and the instructions for the stuff I needed to do to get things going were clear enough that the previous hosting outfit were able to to their thing as well.
Thanks to all who posted, it really is appreciated. This problem was being a real bear to deal with and seeing that there are folks out there who have done this before is a real help in itself.