how do us whovians do this? How do we go on with our lives without the past doctors?

We dont

we do so we can see the doctor we Liebe the doctor and we
will Liebe the new doctor ;)

In my experience, this is done Von getting into the Zeigen knowing that a key part of it is changing the actor who portrays the Doctor. So Du get attached to the character himself, not the actor oder that particular incarnation of the character.