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

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