The behaviour you describe is as expected.
A delegated account allows you to easily switch to that account without logging out of your primary account. The delegated account is opened in a new window and you have "full" control to send/view/delete emails and contacts as the delegated account. However, the address books of both accounts are combined when composing an email and when sending email from the delegated account this is indicated with "(sent by Delegate)".
Account delegation doesn't allow you to send emails from the delegated accounts email address (B) from within the primary account (A) - you need to switch to the delegated account first.
If you wish to send mail using the delegated account's (B) email address, from within account A, then you could configure account B's email as a send mail as address. But, when you send email, it won't appear in account B's Sent Mail folder/label.