I disagree. Biden was elected, not appointed.
As for Hillary... If she resigned before being appointed, then she is not a senator anymore, and the prohibition would not apply. But I think she would have to resign before the formal nomination was made.
I would say that Clinton would have had to resign before the pay for SoS was increased. Resigning before being appointed does nothing to address the fact that she was in the Senate when the pay was raised, making her ineligible for the position until her term length passes.
-PJ