There are free services that allow you to create countless emails, one per site is ideal, just like one (different) password per site. Addy and Simplelogin have a generous free tier, last I checked the first one allows for unlimited receive-only addresses (when shopping it’s very rare you need to respond), the second gives you some two-way addresses.
If you get a domain, many registrars include free mail service, and have mail forwarding, or “redirecting”, which basically will allow you to create countless addresses (that can also send/respond) for your one account (You add these “email forwards”, or “Identities”, to your app of choice, like K9-Mail for android). You don’t necessarily need to buy their separate email package (although the interface might be more convenient). I’ll give you one example which includes email: OVHcloud, one of the largest clouds in europe.
If you can afford it there are all-in-one services like Soverin with easier interface.
It might be wise to start a slow process of migrating (or maybe deleting and creating again) accounts, and saving all this stuff in a password manager (like KeepassXC) if you aren’t already.
Most people give their real full name, phone number and email for any loyalty card wothout batting an eye, plus even with anonymized data it’s useful to the owners to track correlation of purchases, time, location. Definitively what you said too, we all make mistakes (some more than others), every needless complication of a system is a disadvantage to the customer.