What was the point of the scenes with Marian and the cop?

None of it made sense.

To begin with, why was he so paranoid about her? She told him that she fell asleep on the side of the road because she was tired, and didn't want to get in an accident. So why did he act like she committed a crime?

Also, why did she act so sketchy when he started acting questions? She should have just played it cool and pretended like everything was fine. After all, if he knew she stole some money, he would have mentioned it right off the bat.

Then why does he follow her to the used car lot? It's as if he's stalking her.

Speaking of the used car lot, AGAIN, why did she act so sketchy with "California Charlie?" She could have just played it cool, but instead she started acting all paranoid, and made him suspicious.

And since the cop followed her to the car lot, why did she even bother trading in her car? He saw the new car that she bought, so it would have been easy to track her down. Plus, California Charlie had all of her personal information. Trading in the car was useless.

Finally, the cop saw that she was acting very strange, being in a hurry, trading in her car, and they rushing off the car lot.

I don't know how it was in 1960, but any smart cop would have followed her. Instead, he just let her drive off.

Overall, the interaction with the cop had no relevance to the rest of the movie. It didn't affect the outcome, and none of what transpired between them, had any bearing on what happened to her later in the movie, nor did it help provide any clues to Marian's disappearance.

So really, what was the point?