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

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