The safest thing would be to create the new mailbox (firstname.lastname@example.org), verify that it was working, and then switch the alias to forward to the mailbox.
Alternatively, you could setup a catch-all address temporarily. Then when you delete email@example.com, any mail that might get sent to that address would be received by the catch-all instead. You can then forward it to firstname.lastname@example.org when the switch is done.
Whether deleting and re-adding the address as a full mailbox would lose mail depends on how DH operates behind the scenes. It's possible that DH support could do this in a way that doesn't lose mail.