Magento 2.1 Configuration – Why Can’t Change Catalog Price Scope to Website

First, I would do what Sanders said and change your price scope to "Global". Then, if for whatever reason you have website prices that persists, I would run the following query, which will return all of the non global prices saved on your database.

SELECT d.entity_id AS product_id,d.value AS price
FROM catalog_product_entity_decimal AS d
    INNER JOIN `eav_attribute` AS a
        ON d.`attribute_id` = a.`attribute_id`
WHERE a.`attribute_code` = 'price'
    AND d.store_id != 0

First examine these records, if any. If you're seeing website prices, this query should return some results. If you remove these records, all prices will fall back to the global price (store_id = 0). Proceed with caution and always test it on a dev environment first!

Load is pretty expensive memory wise.
You can use this for a faster update:

Mage::getSingleton('catalog/product_action')->updateAttributes(
    array($productId),
    array('name' => $storeSpecificName),
    $storeId
);