I fixed the searches so they just use a page refresh. The AJAX was getting too complicated.
@girl.mg – The simple way to do that is to have two select boxes with the categories in it where the values are the ids. Like so:
<form>
<select name=id1>
<option value=catId>CatName</option>
<option value=catId>CatName</option>
</select>
<select name=id2>
<option value=catId>CatName</option>
<option value=catId>CatName</option>
</select>
<input type=submit value="Go!"/>
</form>
Where each instance of CatName is a category name and catID is the ID for that category.
Next, handle them in your PHP as described here:
https://codex.www.ads-software.com/Template_Tags/query_posts
Which should look like this:
$id1 = $_REQUEST["nameoffirstselectbox"];
$id2 = $_REQUEST["nameofsecondselectbox"]
query_posts(array('category__and' => array($id1,$id2)));
Then the normal loop will print the results.