There's some of that throughout many of the books. Harry never met a woman he didn't like to ogle and describe in intimate, awkward detail. Nor does he ever miss a chance to go all "M'lady" when the opportunity arises. The Molly stuff is definitely the most cringe and, quite frankly, pretty gross though.

It's been a while since I've read them, but you can pretty much just skip the first two books if you want. Other than a few references/characters (that Butcher typically explains/introduces again anyway) I don't really think there's a whole lot of overarching plot introduced yet in those two.

Yeah, he never does quite lose all of the neckbeardy cringe moments, but they become a little less over the top as the books go on.