Every option I tried, both posted hints and third party commercial programs that claimed they would do it, timed out or failed.
The answer was too easy, but it took me three days to figure it out. Hopefully this posting will save someone that time.
phpmyadmin's newer versions have an EXPORT function on the main page. Click on that. On your old server, select the databases you wish to move by highlighting them (or select ALL). Under DATA choose COMPLETE INSERTS. Click SAVE AS FILE and click GO. You will then be asked to load the file to your hard disk. Choose an easy to find folder and download the .sql file.
If your file is very large I suggest making several files, each less than 250,000Kb. (phpmyadmin doesn't have download restrictions but it does have upload restrictions, plus it is limted to the transaction times set in PHP)
Use an FTP program, like CoffeeCup DirectFTP, to upload the resulting file to your new server.
Use an SSH program to get root access to your server. I used Putty. Sign in for root access. Then go to the mysql server with the command
mysql -h localhost
Then type:
mysql < and add the path to the file you uploaded. e.g.
mysql < /home/my_Servers_User_Name/public_html/the_name_of_my_file.sql
BANG! Within a couple of minutes you are up and running.
Good Luck!
Brian Irwin
www.pointsplan.com