I think that the problem might be with the method of the search form. Rite now you have it set to get like this:
<form id="WADASearchForm" name="WADASearchForm" method="get" action="stateoccvisajoin_Results.php">
I think if you updated the code to use the get then it should work for you. The code would look like this with the update:
$WADbSearch1->addComparisonFromEdit("StateOccVisaID","S_StateOccVisaID",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","=",1);
$WADbSearch1->addComparisonFromEdit("ASCO","S_SOVOccupationID",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","Begins With",0);
$WADbSearch1->addComparisonFromEdit("SOVStateID","S_SOVStateID",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","=",1);
$WADbSearch1->addComparisonFromCheck("V176","176","Yes","",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","=","=",0);
$WADbSearch1->addComparisonFromCheck("V886","886","Yes","",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","=","=",0);
$WADbSearch1->addComparisonFromCheck("V475","475","Yes","",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","=","=",0);
$WADbSearch1->addComparisonFromCheck("V487","487","Yes","",(isset($_GET['All']) && $_GET['All'] == 'Yes')?"OR":"AND","=","=",0);
Please give this a try and let us know how this works for you.