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

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