I solved my own issue using get_posts for two different loops:
<!-- IF status is "published" list here -->
<?php
$args = array( 'post_type' => 'publication', 'meta_value'=> 'published', 'orderby' => 'title' );
$published_publications = get_posts( $args );
echo '<h3>Published</h3>';
foreach ($published_publications as $post) : setup_postdata($post); ?>
<div class="publication">
<h4 class="publication_title"><?php single_post_title(); ?></h4>
<?php echo get_post_meta(get_the_id(), 'publication_apa_citation', true); ?>
<?php if ( get_post_meta(get_the_id(), 'publication_online_access', true)): ?>
<h4 class="publication_online_access">Online Access:</h4>
<p><a href="<?php echo get_post_meta(get_the_id(), 'publication_online_access', true); ?>"><?php echo get_post_meta(get_the_id(), 'publication_online_access', true); ?></a></p>
<?php endif; ?>
<?php if ( get_post_meta(get_the_id(), 'publication_abstract', true)): ?>
<h4 class="publication_abstract">Abstract:</h4>
<p><?php echo get_post_meta(get_the_id(), 'publication_abstract', true); ?></p>
<?php endif; ?>
</div><!-- .publication -->
<?php endforeach; ?>
<!-- IF status is "accepted" list here -->
<?php
$args = array( 'post_type' => 'publication', 'meta_value'=> 'accepted', 'orderby' => 'title' );
$published_publications = get_posts( $args );
echo '<h3>Accepted</h3>';
foreach ($published_publications as $post) : setup_postdata($post); ?>
<div class="publication">
<h4 class="publication_title"><?php single_post_title(); ?></h4>
<?php echo get_post_meta(get_the_id(), 'publication_apa_citation', true); ?>
<?php if ( get_post_meta(get_the_id(), 'publication_online_access', true)): ?>
<h4 class="publication_online_access">Online Access:</h4>
<p><a href="<?php echo get_post_meta(get_the_id(), 'publication_online_access', true); ?>"><?php echo get_post_meta(get_the_id(), 'publication_online_access', true); ?></a></p>
<?php endif; ?>
<?php if ( get_post_meta(get_the_id(), 'publication_abstract', true)): ?>
<h4 class="publication_abstract">Abstract:</h4>
<p><?php echo get_post_meta(get_the_id(), 'publication_abstract', true); ?></p>
<?php endif; ?>
</div><!-- .publication -->
<?php endforeach; ?>