>Sell this off and put in a little more money and get a new car.

Does "put in" work here?

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Yeah that works.

Honestly, it would sound normal in spoken conversation, because intonation will help a bit. But especially for written English, having that many "and"s in a sentence can make it get a little awkward and complicated quickly. So if you can rework it to remove one of the "and"s, then that would read better.

So maybe: "Sell this off and put in a little more money TO get a new car."
-- Changing the second 'and' into 'to' will work in this case, and it even adds a bit more clarity to the sentence. So that's a bonus.

That is all assuming that the selling off and then adding some more money is to then be able to get/afford the new car.

If it is just three unrelated tasks, then: "Sell this off, put in a little more money, and get a new car." would be more correct to make it into a list.