Casual is a great one, and has the same problem.
I can’t tell what you’re trying to get at in your second example – is it “skosh”?
The short form of “vegetarian” is close to what the OP asks, although we’ve all kind of collectively agreed that it’s spelled “veggie”. But that’s clearly incorrect following standard pronunciation rules, in which the double-g is pronounced with a hard /g/ sound.
A single-g “vegie” suggests a long /ē/ sound, so that doesn’t work. We could have taken the “fridge” route and gone with “vedgie”, but that never took off, probably because it looks like wedgie.
I cannot think of a good phonetic way to spell it, but I don’t think I’m hallucinating having heard the word. Pronounced, it sounds sort of like ‘shush’ except replacing the ‘sh’ sounds at both ends with a soft ‘J’.
My friend Reggie would disagree 
Even better is “veg” as in to “veg out”, but at least we’ve agree on the spelling. I can at least imagine “veggie” being pronounced like “vedgie”. Cas(ual) and Us(ual) don’t even have that.
ETA: @solost, it might be a regionalism that I’ve never heard.
I also have that (or a similar) tonal series, and I’ve been delighted to observe that one of my kids has taken it on as well.
Found it! I’ve only ever heard it said, so I kept guessing at spellings and Google had no idea until I hit on it by trial and error:
That’s not a term I’ve heard. Looks like it has the same sound as uszh/yushze/whatever.
Yep, except it’s a soft ‘u’. As in ‘put’.
Ah, I missed that - got it, thanks.
The primary focus of this thread seems to be the soft (“voiced”) “sh” sound, which occurs in established English words, but is not very common AFAIK. It’s a known consonant but without a letter for it in the alphabet.
It is most often spelled with “su” as in usual, casual, treasure, pleasure. There’s also “azure” I think. The spelling “zh” is also well-known, but only for spelling foreign words or names, especially Russian (Zhivago, Brezhnev). I’m not aware of any established English words spelled with “zh”.
I guess – I started with yuzh and people added cazh and zhuzh. However, it’s certainly possible that there are other spoken words that don’t have a spelling, although other than zh sounds, I can’t think of any.
The gutteral “ch” sound, as in the German “ach” is somewhat known among English-speakers, although I’ve encountered people who could not pronounce it. I’m not aware of any established English words that have that sound, nor any way to spell it in English. It occurs in some colloquial interjections, in particular “yecch!” in reference to bad-tasting food and similar.
ETA: It’s a very common sound in Hebrew – so common that there are two letters in the alphabet for it!
English and Arabic have some consonants that don’t occur in Hebrew, yet are well-known to Hebrew speakers because of their prevalence in English and Arabic. These include the English “j” sound, the English “ch” sound, and one other I’m not thinking of right now. They are used for foreign words and names, and spelled using similar Hebrew letters followed by an apostrophe-like character.
Is it accurate to say that the shortened version of “usual” rhymes with “luge”?
I think so, but not with yuge, which has a harder G, like huge.
I have never in my life heard somebody shorten “usual”. I expect it’s a regionalism, but I wonder from where? Or else something so far confined to the younger cohorts I don’t have much contact with. Not that my experience is anything but one guy’s anecdote, but the OP’s example leaves me completely baffled.
I’ve certainly heard, and said many times, the shortened form of “casual” that might be spelled something like cahj or cazh as somebody said upthread.
/juːʒ/ (OK, I’m cheating…I’ll go with “yoozh” but that kind of defeats the point of shortening it) is a pretty common spoken shortening in my experience. I’m 46, Chicago, for a couple of data points. I don’t know when I first heard it, but I feel like it’s fairly old (like at least a decade or two) by now.
ETA: Urbandictionary has an entry for “uzhe” that goes back to October 2007, and one for “uzh” in 2008. I checked some other spellings and those seem the earliest entries.
I don’t think I’ve ever heard actually anyone say the shortened form of “usual,” though “yoozh” seems pretty accurate.
“Zhuzh” was pretty popular for a while from “Queer Eye for the Straight Guy.”
55, NYC and NJ.
You mean غ? Note the dot, which makes not correspond to a Hebrew letter, though I suppose you could write it as an “ayin” plus an apostrophe. According to Wikipedia, they do usually try to spell it as “g” or “gh” (e.g. Gaza)
G as in “Los Angeles” ? 
A common misconception (and genuine accent in Hebrew, i.e. some Hebrew speakers pronounce them the same); there are two letters for two different fricatives, only one of which is guttural; the other is uvular.