Don't kid yourself with premium plans... It's still GoDaddy's servers... Just keep that in mind.
Transfers should not have any affect on your rankings, as long as it's an improvement. If you transfer to a bad host, get a bad IP address and actually slow your site down (degrade), of course that would affect your rankings. But, if you upgrade to a better host, you're only risking even better search engine results. I'd take that kind of risk anytime. 
If you want to disable search in description in the header bar, try the following: find your includes/templates/YOUR_TEMPLATE/sideboxes/tpl_search_header.php and change
Code:
$content .= zen_draw_hidden_field('search_in_description', '1') . zen_hide_session_id();
to
Code:
$content .= zen_draw_hidden_field('search_in_description', '0') . zen_hide_session_id();
Of course, if the file isn't there, it means you don't have an override so copy the file from template_default directory to your template's directory.
This is all under the assumption your files haven't been modified and the header search bar wasn't hard coded or in any other way altered.
To test if you actually did something, try searching for a term that can only be found in your product descriptions - if no matches are found, it means you disabled searching in description.