The vibe is get from “ex-STRAW-dinary” is the pretentious sassiety guy saying it. This kind of guy:
https://upload.wikimedia.org/wikipedia/en/4/4d/Original_New_Yorker_cover.png
I economize at the other end, with /strord-nurry", kind of rhymes with “ornery”. British/Canadian style.
Ever reflect on “comfortable”, in which we pronounce the T before the R.
I think it’s time to take all those unnecessary B’s out of “probly”
In many American English dialects (including mine, and probably the ones most commonly spoken), “core” and “straw” aren’t even approximate rhymes. For me, they have very distinct vowellings, and the terminal sound is completely different. For me, “core” has a definite terminal “r”, and “straw” doesn’t.
As discussed upthread, some dialects (yours appears to be one) are non-rhotic. However, as this thread is discussing, the word “extraordinary” seems to increasingly have a non-rhotic pronunciation even in American English dialects that are generally rhotic.
David Axelrod is the worst offender. Officially. An otherwise very smart and articulate guy but he has a mental block on this one (does he spell it “extrodinary”?). And don’t get me started on nucular and realator.
It’s nothing to do with smart or not smart.
There are multiple ways of pronouncing many words in English, and can include pronunciations that have one or more silent letters. (Which isn’t even the case here, because “extrodinary” would be “extr-oh-dinary”. In “ex-straw-dinary” the R is still functioning to denote the “aw” / “or” vowel sound).
Now it may be the case that the rhotic pronunciation of “extraordinary” is more common in the US, but from the other posts in this thread it seems far from universal.
It’s clearly a compounding of extra+ordinary and “odd-inary” is not an accepted alternative to “oRdinary”. Libary and Febuary are also not accepted as grown-up pronunciations.
The English language does not have a committee of Immortals that dictate proper usage and pronunciations. Worcestershire is clearly a compound of Worcester and -shire, yet it’s pronounced, roughly, “wooster”. Wednesday is “wensday”, not wed-nes-day, much less Woden’s Day. Sounds get elided in English compound words all the time, and pronunciation varies between dialects and over time. And, just by the way, “odd-inary” is, in fact, a perfectly cromulent pronunciation in many non-rhotic dialects.
No one is pronouncing it odd-inary. That would indeed be odd.
I pronounce it awed-inary, since sore and saw, father and farther, sauce and source are homophones, whereas cot and caught, odd and awed are pronounced completely differently. 
In my part of the US (in the uppermost northeast) they do not remotely rhyme. I can’t even fathom how someone might attempt to rhyme them they sound so different. Do door and draw rhyme for you? Because for me core and door rhyme closely as do draw and straw.
Well it’s awkward to say if they rhyme because they have the same initial. It’s like asking if cool and cruel rhyme.
But if you change it to “Do roar and draw rhyme for you?” the answer is yes.
If anyone can find a good link with the North East US pronunciations, I’d appreciate it. To me, the words rhyme, and I watch plenty of US TV and I never noticed anything different in the pronunciation of “straw”.
Yeah…roar and draw don’t remotely rhyme either.
As for your question, you need to watch some New Hampshire Chronicle with Fritz Wetherbee.
This is fascinating because two posters have said they “can’t fathom” my pronunciation, but I don’t think I’ve heard theirs either.
Question: In my dialect, the words pause, paws and pores are pronounced the same. In yours, which is the odd one out (or indeed, do all 3 have a unique pronunciation)?
Also pours 
The former is largely true, but the latter is false. FEB-yoo-air-ee is the most common pronunciation of February in the US, at least.
In my dialect, “pause” and “paws” are identical, while “pores” is a completely different word, with different voweling and a completely different terminal sound - the “r” is pronounced.
Yup, pause and paws are identical and sound nothing like pores. And as Mijin said pours and pores are identical to each other.
Is this video on the difference between US / UK / Australia pronunciation of the word “core” accurate?
If so, then I don’t think the difference is as big as some here have suggested, but anyway, it answers the OP; it’s just different dialects of English (and it’s not unusual for a British pronunciation to infect the US, or vice-versa). And both ways of reading the word are pronouncing the “R”, it’s just that with the British way, the R is helping to code for a particular vowel sound, like the difference between “cat” and “cart”.
This makes me think of the Thai lady’s name Porn. Many a guy has been disheartened to learn it’s actually pronounced like the English word “pawn.”
Not sure if this is an intentional joke, but as implied by my link above, “pawn” and “porn” are pronounced exactly the same in many, perhaps most, dialects of English. So it should be many an American guy is disheartened. 
I think I’ve told the story here about the time my English bf and I had a 15 minute conversation about the porn stores and whether they were open on Sundays. He was actually talking about pawn stores.