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

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