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

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