Well, as promised, here's a simple page using your code:
<%@ Page Language="VB" %>
<script runat="server">
Private Sub btnGo_Click(ByVal Sender As Object, ByVal e As EventArgs)
litX.Text = "You selected " & txtg4action.Text
End Sub
</script>
<html>
<head>
<title></title>
<script type="text/javascript">
function ddlChange() {
var ddl = document.getElementById('<%=DDList.ClientID %>');
var textBox = document.getElementById('<%= txtg4action.ClientID%>');
textBox.value = ddl.options[ddl.selectedIndex].value;
}
</script>
</head>
<body>
<form id="Form1" runat="server">
<p><asp:DropDownList runat="server" ID="DDList" onchange="ddlChange();">
<asp:ListItem Value="1" Text="Option 1"></asp:ListItem>
<asp:ListItem Value="2" Text="Option 2"></asp:ListItem>
<asp:ListItem Value="3" Text="Option 3"></asp:ListItem>
<asp:ListItem Value="4" Text="Option 4"></asp:ListItem>
</asp:DropDownList></p>
<p><asp:TextBox runat="server" id="txtg4action"></asp:TextBox></p>
<p><asp:Button runat="server" ID="btnGo" OnClick="btnGo_Click" Text="Submit" /></p>
<p><asp:Literal runat="server" id="litX" /></p>
</form>
</body>
</html>