@Milo and @Howdy_McGee had the answer.
I found the section I needed to modify in my functions.php file:
case 'text':
echo '<input type="text" name="', $field['id'], '" id="', $field['id'], '"
value="', $meta ? $meta : $field['std'], '" size="30" style="width:97%" />',
It was the value field that was getting truncated, so I passed it through esc_attr() as suggested:
case 'text':
echo '<input type="text" name="', $field['id'], '" id="', $field['id'], '"
value="', esc_attr($meta ? $meta : $field['std']), '" size="30" style="width:97%" />',
Problem solved.