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 door getting into the toon knowing that a key part of it is changing the actor who portrays the Doctor. So u get attached to the character himself, not the actor of that particular incarnation of the character.