account names cannot be changed.
you can only change your display name, which is available in the settings.
whether display names or usernames are shown depends on the interface/client and user settings where available.
the only way to change the username is to create a new account.
verification emails are usually sent immediately. if there are delays you should check your junk folder, and if it’s not there it probably won’t arrive anymore. depending on the instance you signed up on there may be alternative methods to reach out to the instance admins about this. note that private messages from mastodon to lemmy do not work unfortunately.