I was able to do this using Albert Chu’s and petersv’s comments as guides.
A notable point of confusion is that one must access the “My Logins” link from a Stack Exchange community site like Graphic Design Stack Exchange, Stack Overflow, or Super User to delete an old email address, not https://stackexchange.com/ where one would most expect to be able to edit their account. There’s no “remove” link if you access “My Logins” from the core site.
Furthermore, after deleting an old email address, the associated Stack Exchange OpenID account and its vanity URL still exist at https://openid.stackexchange.com/, albeit separate from the new account. If you wish to assign that URL to the new account, you’ll need to log into the old account and delete the vanity URL, and then log into the new account and assign the released URL to it.
This also means you’ll have a dangling account each time you need to change your email address since the old accounts apparently can’t be deleted.