I think you load the file ajaxSearch.js twice. This is the reason why you get the close icon is attached twice to the form.
The js source code is the following:
sf = $(p+'ajaxSearch_form');
if (sf) {
var c = new Element('img'); //= Close Image
c.setProperties({src: _close, alt: _closeAlt, id: p+'searchClose'});
c.setStyle('opacity', '0');
sf.appendChild(c);
as['sc'] = c;
Where is your snippet call ? on your template or on your page ?
Could you post your AS snippet call. Thanks.