12 Angry Men is pretty stiff and theatrical. Not a criticism since it is a great movie, but it it wolds apart from say Five Easy Pieces.
I've only seen Breathless and A Band Apart. My sense is that Godard played a big part in transforming cinema from mannered to free-form. But I'd caution against framing that transition in terms of quality. Lots of stiff, theatrical movies are excellent, many of them have really good camerawork, and Godard certainly didn't invent moral ambiguity.