Connecting Cart API IDs for permalink cart abandonment

Shopify Partner
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Current I am using a separate email platform that will continue to send out our abandon cart emails. But we are using a home grown, built from scratch website that has been migrated to Shopify. I have been able to access the Cart.js but can't seem to connect any of the cookies within that file to bring back the cart on another browser. I have tried the cart cookie, the cart_sig, cart_ts, _shopify_y, but nothing recalls the cart which is weird because I am using my own checkout and that can call the cart API using the cart cookie to pull in the cart contents. Has anyone figured out a way through the URL to call back the cart across browsers or a work around like checking to see if they are on the same browser and if not then adding the contents to a new cart? Any help here would be greatly appreciated.