Multiple Custom Taxonomy Dropdowns Lists

I have 3 hierarchical (category-like) custom taxonomies in a WP site, books, characters and writers. I want to create a dynamically populated dropdown menu of terms for each of these in the site's sidebar, so that on selecting a term in any of the dropdown the site automatically loads the corresponding archive.

Using the method described here I can get this working on one taxonomy only. As soon as I add more I find that only the last one works.

Here's what I have so far. Clearly I need to find a way for later instances of the function to not over-ride the earlier ones. I thought that changing the IDs and function names would do this but it's clearly not that simple.

?php
    $writers = get_categories('taxonomy=writers');
    $select = select name='writers' id='writers' class='postform'n;
    $select.= option value='-1'Writers/optionn;
    foreach($writers as $writer){
        if($writer-count  0){
            $select.= option value='.$writer-slug.'.$writer-name./option;
        }
    }
    $select.= /select;
    echo $select;
?
?php
    $characters = get_categories('taxonomy=characters');
    $select = select name='characters' id='characters' class='postform'n;
    $select.= option value='-1'Characters/optionn;
    foreach($characters as $character){
        if($character-count  0){
            $select.= option value='.$character-slug.'.$character-name./option;
        }
    }
    $select.= /select;
    echo $select;
?
?php
    $books = get_categories('taxonomy=books');
    $select = select name='books' id='books' class='postform'n;
    $select.= option value='-1'Books/optionn;
    foreach($books as $book){
        if($book-count  0){
            $select.= option value='.$book-slug.'.$book-name./option;
        }
    }
    $select.= /select;
    echo $select;
?

JS:

script
    var dropdown = document.getElementById(writers);
    function onWriterChange() {
        if ( dropdown.options[dropdown.selectedIndex].value != -1 ) {
        location.href = ?php echo home_url();?/writer/+dropdown.options[dropdown.selectedIndex].value+/;
        }
    }
    dropdown.onchange = onWriterChange;

    var dropdown = document.getElementById(characters);
    function onCharacterChange() {
        if ( dropdown.options[dropdown.selectedIndex].value != -1 ) {
        location.href = ?php echo home_url();?/character/+dropdown.options[dropdown.selectedIndex].value+/;
        }
    }
    dropdown.onchange = onCharacterChange;

    var dropdown = document.getElementById(books);
    function onBookChange() {
        if ( dropdown.options[dropdown.selectedIndex].value != -1 ) {
        location.href = ?php echo home_url();?/book/+dropdown.options[dropdown.selectedIndex].value+/;
        }
    }
    dropdown.onchange = onBookChange;
/script

Many thanks for any help.