Shopping cart empty after moving to a new web host

Usually this is an issue in the php.in with the session_save_path being set to a directory that either doesn't exist or doesn't have proper privileges. This is probably something your hosting provider would have to address for you.

If you create one page that has:
<?php
@session_start();
$_SESSION["test"] = "works";
?>

and then create another page with the code:
<?php
@session_start();
var_dump($_SESSION);
?>

Then visit both pages on your site. If sessions are working then the second page will have the "test"="works" session variable listed. If it isn't then you can use those pages to demonstrate the issue to your hosting provider so that they can fix it for you.