I hope a reference to a service is acceptable here. I’ve been using SneakEmail.com for years. It’s a for-fee service (quite reasonable) which allows you to create any number of email addresses for use everywhere. I have a different address for every website and company that I contact. If they divulge my address, I know exactly who did it, and I can disable or delete the address. You can also create addresses on the fly, like when you walk into a store and need to give an address to get some discount or join a club.
BTW, I came here looking for the same thing as the OP. I really want to be able to generate and maintain email addresses for my DH domains via the API. This topic has been brought up many times in the past and I’d really appreciate it if DH would add the feature. Honestly, of everything going on at DH, this is the only feature that I actually want from them outside of what we have.