I played around some more and got the fields to appear, but then after clicking Continue, noticed the new info wasn't actually added to the order. Instead of spending even more time on this, I realized using text attribute fields on some product pages was a better way to get the info I needed.

In case someone else looks to solve the original question, adding the following to tpl_modules_checkout_new_address made the extra fields appear:
PHP Code:
<label class="inputLabel" for="extrafield"><?php echo ENTRY_EXTRAFIELD?></label>
<?php echo zen_draw_input_field('extrafield'''zen_set_field_length(TABLE_CUSTOMERS'customers_extrafield''40') . ' id="extrafield"') . (zen_not_null(ENTRY_EXTRAFIELD_TEXT) ? '<span class="alert">' ENTRY_EXTRAFIELD_TEXT '</span>'''); ?>
<br class="clearBoth" />
...
<label class="inputLabel" for="extrafield4"><?php echo ENTRY_EXTRAFIELD4?></label>
<?php echo zen_draw_textarea_field('extrafield4''30','2'$extrafield4'id="extrafield4"') . (zen_not_null(ENTRY_EXTRAFIELD4_TEXT) ? '<span class="alert">' ENTRY_EXTRAFIELD4_TEXT '</span>'''); ?>
<br class="clearBoth" />
My fields weren't required so I skipped the code of "<?php
if (xx == 'true') ..."