Gonna be honest. I seriously have a suspicion that Cortana is actually hiding in plain sight. She has either deleted "The Weapon" and replaced her and is playing coy.

Or she injected herself into New Cortana and over time will come back piece by piece.

The cutscene we saw in the E3 Story trailer is obviously later on in the game. John already had a conversation with Original Cortana at some point where she says "Theres not much time." "If you knew how you were going to die. How would you live your life?"

Then there is another time jump to his conversation with The Weapon.