It does not really matter what PROCESS you have to use to "export" the dbase.
It seems (from your posting) that what happens is that when you've gone through all this, you are being confronted with a screen of text that you are now being told to SAVE AS a sql file.
Just do this...
Get CRIMSON EDITOR off the web - it's a FREE text editor, and anyone working with php, sql and html ought to have it (or something like it)
Anyway, get CRIMSON EDITOR working on your computer.
Open up Crimson Editor, and start a new file (just like you would in MS Word.)
Then, when you're confronted with that block of text from your database (also known as a database dump), place your cursor right at the start of it.
Hold down your SHIFT key..
Scroll to the bottom of that big text block..
Put your cursor right at the end of it...
There will be a final " ; " which you must include...
COPY (ctrl+c)
Then PASTE (ctrl+v) into your new crimson editor file
Choose SAVE AS in crimson editor and select the SQL option from the file type
Call the file " my_database_backup_15_march_2007.sql "
Save it to a folder called DATABASE BACKUP
(This SQL file is little more than a simple TXT file, but by giving it the SQL extension, crimson editor will recognise it as sql and display it in the proper manner).
You now have your database all sewn up, on your hard drive!
FTP all your zencart folders FROM your host, TO your Hard Drive.
You now have a copy of your webshop on your hard drive.
Get the sql file and the zencart folders/files to your new host - either by ftp, or by putting them on a disk and popping it in the post
Ideally, you should do a fresh install at your new host, and there are instructions elsewhere in this forum for doing that and dumping your SQL file into the new database.
But you should consider hosting with a certified zencart host... They will help you get the new site up and running.