So which pronunciation is standard for the [ʊ] sound? Rounded or unrounded?
Certainly there is some rounding, but because roundedness is not phonemic in this position, there is also considerable variation in how much of it actually occurs in any given word and speaker.
For example, you will find that it is generally somewhat more rounded in pull and full than it is in put and foot respectively. That’s because having an r or an l right next to it rounds it off a bit — which is why it is a bit more rounded in root and rook than it is foot or cook. Same with rookie versus cookie, where the first version is a bit more rounded than the second. And of course, a w helps: compare how wool is even more rounded than full, and also moreso that wood.
I believe English has no words with [ʊw], as that seems redundant. However, it can occur in phrases, especially in some dialects, where something like I knew it full-well may approach that.
However, it is still perceived as the very same phoneme in all those words and cases I’ve just listed above.
Correction — or not
I said that I thought English had no words with [ʊw] in them. And at the end of the day, I still believe that. However, I have discovered that grepping the OED yields the apparent existence-proof counterexample of Rauwiloid, which means:
A proprietary name for a hypotensive preparation containing a number of alkaloids extracted from Rauvolfia serpentina.
You also have compound words whose first element ends in [aʊ] (rather than [aw], as it is sometimes spelled) connecting to something that begins with [w], and which have in effect a “double w” in them, you expand the list to include such things as:
bow-wow, powwow, skeow-ways, wow-wow
Finally, if you consider the sound in words like no and micro to be
an [oʊ] diphthong rather than [ow], then you get all these, most of which were originally compounds of some sort:
froward, frowardly, frowardness, glow-worm, Holloway,
hollowwort, Howeitat, Khowar, meadow-wink, microwave, microweld,
Moldo-Wallachian, nowise, Oldowan, Parowax, powan,
shalloway, slow-worm, swallowwort, werowance,
yellow-wood, yeowoman.
For example, yeowoman theoretically yields /ˈjoʊwʊmən/, at least in North America. Still, there is a reasonably convincing argument to be made that that one is better written as simply /ˈjowʊmən/.
Slightly less uncommon is nowise, which is a compound of one word ending in a diphthong connected to another starting with a triphthong, so /ˈnoʊˌwaɪz/.
But I am still highly dubious of the existence of [ʊw], because I think it fuses into the semi-consonantal glide, [w]. After all, nowise and no eyes are homophonic, so I think this idea of [ʊw] is very hard to justify, and so I stand by my initial statement.
Even towel is usually pronounced with just one syllable, /taʊl/, thereby rhyming with cowl /kaʊl/. Even with folks who work very hard to put two syllables into that, with /ˈtaʊ.wəl/, I submit that you could write that /ˈtawːəl/ and avoid the whole controversy of whether a semi-vowel/semi-consonant/off-glide is really /ʊ/ or really /w/. However you write it, it seems like the same sound to me, such that bisyllabic towel just has a geminate [w]: /ˈtaw.wəl/.
Take a look at the information I link to on this page, which I have titled The geography of America's dialects and cultural affinities compared.
In particular, visit the links for the dialect maps produced by Rick Aschmann, where you will find both audio and video clips illustrating how various sounds are typically pronounced in different regions of the United States. The page depicting his principal dialect map contains numerous supplementary sections that discuss particular linguistic features of American English. Among them is a section on R-coloring which is directly relevant to your question.
I can safely say that both Aschmann's summary of R-coloring and the clips I mentioned will demonstrate that the woman you referred to is thoroughly mistaken in asserting the existence of a so-called 'correct' way to pronounce the sound of the letter R in American English. There are in fact many different ways that Americans sound that letter; it would not be reasonable to demand that they should change their pronunciation merely to suit her preference.
Best Answer
The /t/ in negative contractions in English can have three main realisations. In decreasing order of likelihood (all other things being equal):
It does not matter at all in negative contractions whether the following sound is a consonant or not in terms of dropping the /t/ altogether. It can easily be a vowel that follows.
So in terms of what native speakers actually do, by far the rarest realisation is with a canonical [t]. However, it is never wrong to use a normal [t] sound. Knowing that a [t] will usually not be present will greatly improve non-native speakers' listening skills though.
The other reason to be aware of the fact that there may not be a [t] present is that it enforces that fact that it is stress which is the most important factor for distinguishing negative contractions from normal auxiliaries. Negative contractions are stressed in English, whereas other things being equal most auxiliaries aren't when occurring in positive sentences. So when trying to distinguish between She can come and she can't come we will listen out for the following rhythms:
The first is what we expect from the positive polarity sentence. The second is the negative.
Assimlatory processes
The final [t] in negative contractions may be affected by the sounds following it.
For example, if the word following the contraction normally starts with [j], as in the first sound in you, then the /t/ and the /j/ may coalesce to form an new affricate sound, /tʃ/. This is the first sound that we hear in words like chair. So the string don't you may be realised as:
Also if the following sound is not alveolar, both the /n/ and the /t/ may change their place of articulation according to the place of the following sound. So for example if the following sound is bilabial, the /nt/ cluster may be realised as /mp/. It is quite common to hear RP speakers saying I cam'p believe it, for example.