Why did Skinny tell Badger to ditch his car?

TLDR; To create the impression that he had crossed the border.

The police have a wide presence and a lot of resources when they really want to find someone. Broadcast a request on the network with a basic description (All-Points Bulletin or APB). This won't be a detailed identification, and they'd normally list only the major details ... like the kind of car the suspects are driving. Then the police will actively start searching.

You see a lot of movies where people change their clothes when they are being hunted - because clothes are a much more obvious indicator to look for.

Escape is as much about deception as hiding or speed. These are good for breaking contact - it's easier to find someone if you can see them or their tracks. You have to get them to stop looking, or at least stop actively looking. Hiding the car near the border gives the indication that they have crossed over the border in some other manner, and therefore the suspects are no longer in the police jurisdiction. File the paperwork and go back to a different case.