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 사랑 the doctor and we
will 사랑 the new doctor ;)

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