They can only send the marketing emails if you have the "send me marketing emails" checkbox checked when you type in your email address. Dark pattern, yeah.
Either way, if I enter my e-mail, decide "nah" and quit, that should be that. It shouldn't be sending my details in the background
I use this Shopify feature as a buyer to get discount codes, as abandoned cart emails often come with them.
They require your full address to show you shipping prices as well (vs just letting you estimate with country and postal code alone), so I have to punch in a ton of details before I nope out at awful shipping costs, and I end up with these emails.
You're probably getting the "abandon cart" emails, which is legally considered a "transactional email", and doesn't require consent.