Extraordinary - I always knew it to be pronounced “Eks-tror-din-ary” but at some point I started noticing people pronouncing it as “Extra-Ordinary” Which is the preferred pronunciation?
Often - Half the people I know pronounce it like “Coffin” with the c dropped. Others pronounce it as “off-tin”. I seem to use both at times. Which is more common?
This may depend on what variety of English you speak. Or it may not; in my native HibE, both pronunciations of both words are current, and which you favour is a matter of taste.
The OED confirms both pronunciations of both words, and with respect to “often” confirms both pronunciations for AmE as well as for BrE. It notes Middle English spellings of “often” including offen and offtyn, which suggests that both pronunciations go back to Middle English.
I use “ex-troar-dinary” usually with native-English speakers.
I use “extra-ordinary” at other times.
I use “often like coffin” normally.
I use “of-ten” with Japanese who can’t speak English well.
Try listening to Obama’s political strategist, David Axelrod’s podcast, The Axe Files. He grew up in Brooklyn and “extraordinary” is a word he uses pretty much each episode, and hits with clear glee, accent on full display. Hmm, how does he say it. “Es-trahd-nurri” maybe?!?
Extraordinary - “Eks-tror-din-ary”
Often - “Coffin” with the c dropped.
I don’t use extraordinary often 
but I’m pretty sure I’d pronounce it extra-ordin-airy if it wasn’t immediately attached to a following noun, if it were then it would depend on how much emphasis I was placing on it.
I think I alternate between offen and offten.
Same here. “Often” with the “t” pronounced sticks in my craw a bit, but it seems (from my observation) to have become close to if not the majority pronunciation (and it is logical, of course, being spelled with a t.") Both are acceptable pronunciations, of course, but I had been conditioned a bit by school teachers when I was younger to drop the “t” (so it follows the pattern of words like “soften” and “listen.”)
Now, with “extraordinary,” it’s the five-syllable pronunciation above most of the time, but sometimes, I’ll say it with all six (as “extra” + “ordinary”) when I draw it out for emphasis. (Or, in a much more esoteric situation, when I need to refer to it as the title of a Stephen Schwartz song from Pippin, as it is sung with six syllables in that.)
Extraordinary: (in English) I pronounce it “extroh-dnery” with the “oh” drawn out.
In French, the equivalent adjective is extraordinaire. People from France (AFAIK) pronounce every letter. In Québec, the middle “a” is very short or omitted entirely, so it’s more like “extrordinaire”, “estrordinaire” or even “strordinaire”.
Often: “offen”.
When I hear people say “extra-ordinary” or “of-ten,” I always want to say, “easy there, Frasier.”
“Eks-tror-din-ary” with maybe a little more “air-y” at the end. And as for the other “off-ten” almost as two distinct words/clear syllables.
I would pronounce “Coffin” differently than “Often”, in that the “-in” in Coffin would be a little more “-in”, whereas the “-en” in Often would be a little tighter with a barely noticeable “schwa -n”.
So - Cawfin vs Awfn.
What’s the thinking equivalent of an ear-worm? Brain-worm? For the first time in all these pronunciation threads you started one for me. 
I’ve been thinking and listening to myself and just as I do with “often” I add a little extra to the first – “Eks-stror-din-airy”. I am clearly putting an extra s in there as the end and beginning of distinct syllables.
Originally north-east PA mountains, now southwest PA in case its some sort of regional thing.
No love for “eks schtroard in ree”? Actually I say “ex stroard in airy”.
And “offen” is the way for me. Off ten sounds pretentious.
Totally! I know some people who do this and I know they are trying to sound smart. It doesn’t make them sound smart.
“Ex-tror-dinary” for me. The “extra-ordinary” I don’t think I heard until I was an adult. It sounds put on to me but I guess some people say it naturally.
I say “offen” (like coffin) after having been corrected to do so by an elementary school English teacher. It stuck although hearing other people say “off-ten” doesn’t sound odd to me and I know now that both are acceptable. This might be the first or perhaps only thing that I consciously changed about my pronunciation. I now sound very little like my siblings.
I offin pronounce the “t” in often. I don’t know why I sometimes do and sometimes don’t. It’s rather extrordinary.
I pronounce it ‘ex-strawdinerry’ when talking about something amazing, but sometimes say it ‘extra-ordinary’ if I’m talking about arranging one more meeting than usual, or something equally dull.
I say, Ex-troar-din-ary and off-ten
exTROARdinary
awffehn – different second vowel sound than coffin
Surely not!