As I said, just submit form with JS and then you’d know which form initiated JS code and where to show confirmation.
That’d be my choice and better UX without page reload.
As for sending e-mails – I have not tried the App you’ve mentioned. Often clients have something like Klaviyo which has Flow integration.
Otherwise an App by Shopify https://apps.shopify.com/shopify-email would be my first suggestion (and it integrates with Flow).