You’ll probably want to use get_field instead of the_field in your assignment. the_field will display the retrieved value automatically which is why you’re seeing duplicate output.
<?php
$adkCopyright = get_field( 'adk_copyright', 'option' );
if ( !empty( $adkCopyright ) ) {
echo $adkCopyright;
} else {
echo bloginfo( 'name' );
}
?>
