This question will seem weird to a native speaker because “I better don’t” sounds inherently wrong and unusual. But if you think about it, it’s an irregularity; normally when a verb is negated and there is no auxiliary already present, the auxiliary “do” is inserted and usually contracted with the “not” to form “don’t”. The exceptional construct “I better not” baffles some non-native speakers.
What is the etymological history behind the construction “I better not (+verb)”? Did this sentence use to contain an auxiliary which has disappeared?