This behavior is expected due to the way Shopify themes are constructed. Shopify requires the product price to be configured as its own block within the page, which does not allow for the product price to update dynamically with the app block widgets (Legacy and Subscription Widget) and the Advanced Subscription Widget. 

Recharge adds the Subscribe and Save price to the widget in order to differentiate the two prices. If you would like the price to update you must use the Legacy Javascript Widget.

You may be able to find a theme that allows you to hide the product price block and rely on the subscription widget for pricing. Some Shopify themes allow you to hide the product price on the product page. You can reach out to Shopify Support for more information. 

You may also be able to custom code your theme to hide the price field, however it is important to note that theme customizations like this are not supported by Recharge and it is recommended to work with a third party developer to implement this kind of change.