Auxiliary Verbs – ‘Hadn’t Better’ or ‘Had Better Not’?

If I did not have a car, I would... is correct here. If I had not a car is not, because you aren't negating the right element. "a" is the indefinite article, but it also counts 'one' in a sense. As such, you would need to negate 'a':

If I had a car, I would drive to the store.

becomes

If I had no car, I would walk to the store.

However, in English you would more commonly say "did not have a car", because negating "have" makes more sense, and so "did not have" places the emphasis on that part of the sentence. You could also say "If I did have a car," which would emphasize that also, but we don't tend to in my experience. I'm not entirely sure why the distinction, but "If I had no car" sounds odd to me despite being technically sound, and "If I did have a car" also sounds a little off.

Had better in [had better + bare infinitive] can be seen as a two-word item that functions as a modal verb, meaning essentially should or ought to. (At least in terms of traditional grammar; modern grammars likely better classify and/or describe this.)

What seem like past forms (preterites) in modals do not necessarily confer a past meaning on the infinitives they modify. An example of this is Could you help me? Although could can be used as the past tense of can, it does not indicate any sense of the past in this example.

A deep explanation as to why this is so relies on understanding the evolution of English at a fairly sophisticated level, which is beyond my knowledge.

Should better is not normally used in Standard English, or if it occurs, is probably uttered by some speakers informally and may have arisen as a sort of fusion of should and [had] better.

The following Wikipedia article, in somewhat fragmented form, addresses some of these issues, and includes a subsection specifically on had better and ought to.

https://en.wikipedia.org/wiki/English_modal_verbs