I have two tutorials by Cook and by Cameron. Cameron suggests that these two groups are pronounced differently: e.g. in "drop" the tongue is perfectly flat, but in "jaw" it goes up by 1/8 of an inch. Cook claims there is no difference. Who is right? What is the best way to handle this?
It's actually the same in some parts of the US and different in others. It's a phenomenon known as the cot/caught merger, and it's one of the primary features linguists use to classify American dialects.
Worth noting that in your specific example of "drop" vs. "jaw," there is a marked difference in length, regardless of the merger, due to the placement of the vowel at the end of the word vs. before a stop consonant.
Technically Cameron is right - the two pronunciations are slightly different. However, most speakers won't notice the difference. If you really want to be precise, your mouth should make more of an "O" shape for "drop" and be a little wider with the cheeks pulled back for "jaw". Also keep in mind that even within the United States, pronunciations vary widely.