I think he mistook her interest in the story for an interest in him.

He first claimed he dropped her off near a restaurant, but the restaurant owner had cctv tapes that showed no such thing. That’s likely when police figured it was murder. 

My mom told me that, thankfully.

Here is an idea. A 3 min show every night at midnight, where Letterman heaves an object off the roof of a tall building, just to watch it smash onto the concrete below. He could still have a guest, to choose the object. Seeing how 2018 is shaping up, I think it might work.