Вы не используете ключи, поэтому просто получите текущее значение и затем перейдите к следующему:
foreach($new_id as $new_ids) {
$propertyAddress = PropertyAddressManagement::find($new_ids);
$propertyAddress->address = current($address);
$propertyAddress->city = current($city);
$propertyAddress->state = current($state);
$propertyAddress->publicOnly = isset($_POST['publicOnly']) ? current($publicOnly) : 0;
$propertyAddress->save();
next($address); next($city); next($state); next($publicOnly);
}
Однако, если ключи одинаковы во всем массиве, то я думаю, что на самом делеэто должно работать:
foreach($new_id as $key => $new_ids) {
$propertyAddress = PropertyAddressManagement::find($new_ids);
$propertyAddress->address = $address[$key];
$propertyAddress->city = $city[$key];
$propertyAddress->state = $state[$key];
$propertyAddress->publicOnly = isset($_POST['publicOnly']) ? $publicOnly[$key] : 0;
$propertyAddress->save();
}