Need help with Link Category re-ordering

Still trying to get help with this one. Sorry for the long post, but I want to try to be descriptive as posible.

My site http://www.davidryanpaul.com is a mod of a free template by stacey leung. I’ve got everything I want done, except for re-ordering the sidebar headings. I created 2 new link categories trhough WP admin called ‘artwork’ and ‘links’. I opened up the sidebar.php, but those new entries aren’t listed. I expected that. But what I don’t uderstand, is how to adjust it so that my new link category ‘artwork’ is the 1st heading displayed, right above ‘pages’. I’ll put ‘links’ somewhere else later. I’ve even tried to make new entries of those categories in the sidebar.php but then my page displays 2 ‘artwork’ sections. Remember, I’m new at this, and I’m very confused. Below is the entire sidebar.php code. Can some kind soul show me how do do this or what it should look like *exactly*. I’m but a simple artist, with no coding experience. Your help is much appreciated.

<div class=”rightimage”></div>
<div class=”right”>

<form method=”get” id=”searchform” action=”<?php echo $_SERVER[‘../wp/flowers/simplicity/PHP_SELF’]; ?>”>
<input type=”text” value=”<?php echo wp_specialchars($s, 1); ?>” name=”s” id=”s” size=”23″ />
<input type=”submit” id=”searchsubmit” value=”search” />
</form>

<div class=”sideheading”>pages</div>

<?php wp_list_pages(‘title_li= ‘); ?>

<div class=”sideheading”>categories</div>

<?php wp_list_cats(); ?>

<div class=”sideheading”>archive</div>

<?php wp_get_archives(‘type=monthly’); ?>

<?php
$link_cats = $wpdb->get_results(“SELECT cat_id, cat_name FROM $wpdb->linkcategories”);
foreach ($link_cats as $link_cat) {
?>
<div id=”linkcat-<?php echo $link_cat->cat_id; ?>” class=”sideheading”><?php echo $link_cat->cat_name; ?></div>

<?php wp_get_links($link_cat->cat_id); ?>

<?php } ?>