I’ve divided my answer into three parts, addressing three epochs in the real or imagined life of the words grunge and grungy. In brief, the nineteenth-century usage of grunge seems to have evolved naturally from other, similar dialect words; but it appears to have no obvious connection to or influence on the later manifestations of grunge and grungy. The 1920s usage of grungy is difficult to nail down, since—as far as I can tell—no one on the Internet has yet provided a citation to an instance of that usage in action. And the modern usage of grunge seems to have started a few years earlier than the mid-1960s (the starting time that most references attribute to it), but with no evident connection to the nineteenth-century British usage or to the widely claimed but mysteriously ghostly 1920s-era usage.
Nineteenth-Century Scottish Grunge
The example from Joseph Wright’s English Dialect Dictionary that phenry reproduces in his answer can be found rendered at greater length as item 24 on page 775 of Alexander J. Ellis, “The Lowland Division of English Dialect Districts, Being Chiefly Those Lying in Scotland: The Mid North Lowland; Third Example. The Fight,” in On Early English Pronunciation with Especial Reference to Shakspere and Chaucer (1889), where the author translates the relevant excerpt as follows:
She [Miss Elizabeth Gray] promised first to one, and then to another, to allow them to accompany her home, and when away-going time came, [Jock] Shanks had her at the door setting out, but [James] Bruce was sitting there, and the sly (faithless) coquette took a nip of his whiskers on the going past, and he got up to his feet, and the loving little girl swung herself out of Shanks’ plaid, and clutched Bruce by the arm, and went off with him, and fairly gave the slip to Shanks, and he always had a deep revengeful feeling after it at Bruce.
The last clause of the translation deals with (approximately) this original wording:
en hii a’i ha’d y gruuqsi efter’t) at :bris.
Ellis’s footnote to the translation then observes:
—deep revengeful feeling, the nearest word to grungy in Jamieson is “grunye, promontory, mouth ludicrously, a grunt.” After this Mr. Innes [who provided the examples and translated them] adds, “The narrative here enters on another subject, and I believe you have had more of it than can be of interest to you. I shall there fore not follow it further.”
I get the impression that the version of the sentence in Wright’s dictionary (the one quoted by phenry) is simply another translation of Innes’s original example.
In any event, several nineteenth-century Scottish dictionaries provide interesting coverage of grunge and its possible antecedents. First, from John Jamieson, An Etymological Dictionary of the Scottish Language (1818), we have these entries:
To GROUNCH, GRUNTSCH, v. n. 1. To grunt. Ruddiman. 2. To grumble, S.B. Douglas.
To GRUCH, v. n. To grudge. Wyntown.
To GRUMPH, v. n. To grunt, S.
GRUMPH, s. A grunt, S.
GRUMPHIE, s. A vulgar name for a sow, S. Ramsay.
GRUNYE, s. Promontory.
GRUNYIE, s. The mouth, ludicrously, S.
To GRUNTSCH. V[ariant]. GROUNCH.
A more elaborate description of “grounch, gruntsch” appears in the glossary to J. Sibbald, Chronicle of Scottish Poetry (1805):
Grounch, Gruntsch, to grudge, to murmur, to grumble, to express displeasure by protruding the mouth like the snout of a pig. [A]lso, to dig like a swine.
Jameson, A Dictionary of the Scottish Language (1846) retains most of it predecessor’s entries and definitions, but it also adds these new entries:
To GROUNGE, GRUNGE, v. a. 1. To look sullen or sulky, Roxb. 2. To grumble, to murmur ; as, “He’s aye groungin’ about something,” ibid. This seems nothing more than a provincial variety of Grounch, Gruntsch.
To GRUNGE, v. n. To look sullen. V[ariant of] Grounge.
And finally, in Johnstone, Jamieson’s Dictionary of the Scottish Language (1896), all of the entries remain the same as they were 50 years earlier.
Grunge also shows up in a nineteenth-century publication of the English Dialect Society. From F.M.T. Palgrave, "A List of Words and Phrases in Every-day Use by the Natives of Hetton-Le-Hole, in the County of Durham" (1896):
Grunge. To grunt. 'Grunt' unknown. "they will shew their teeth at you and grunge at you."—Boy's essay.
From these sources, it appears that the nineteenth-century Scottish and English term grunge was closely connected to the grunting and grumbling of pigs, whether by extension from the earlier term grounch, or independently.
Roaring ‘20s Grunge
I couldn’t find any any mention of grunge or grungy in the sense of “enviousness” or “envious” in my Google Books search results. Perhaps more significantly, I couldn’t find any mention at all of grunge or grungy in a Google Books search for the years between 1900 and 1958 (aside from a couple of dictionary entries referring to the older Scottish and English dialect meanings discussed above), suggesting that if grunge or grungy existed as a slang term in the 1920s, the term wasn’t especially common in published work of the period.
Reinforcing that impression is the fact that more-recent compendiums of slang—most notably Chapman & Kipfer, Dictionary of American Slang, Fourth Edition (2007), which is generally conscientious about reporting earlier senses of slang words—begin and end their coverage of grunge and grungy with the modern senses of the terms. Mr. L. Venier of the Dufferin-Peel Catholic District School Board of Mississauga, Ontario, is the source for “grungy” = “envious” in the OP’s citation of 1920s slang, though Mr. Venier seems to have drawn his list verbatim from the 1920s Slang Terms and Definitions page of the Shot in the Dark Mysteries website) or from The Internet Guide to Jazz Age Slang or from one of a number of other echo-chamber pages that dot the Internet.
Unfortunately, each of these sources for 1920s slang usage (except a couple of brave pages that offer “Grungy: jealous”) supplies the brief notation “Grungy: envious” and lets it go at that—no citation, no example from a contemporaneous book, nothing else. I did find a December 8, 1996, article by Howard Goodman of the Philadelphia Inquirer titled “That Groovy Lingo Ain’t So Happenin’ After All,” which interviews an “avid logophile” named Tom Dalzell. Dalzell notes that many supposedly new slang terms actually have had earlier incarnations as slang terms, sometimes with very different meanings. The following text appears about two-thirds of the way through the article:
”The real birth of American youth slang was in the '20s, with the flapper,” Dalzell says, recalling the gin-guzzling, bob-haired, corsetless free spirit of F. Scott Fitzgerald memory.
”In retrospect, it was kind of dopey and very white," Dalzell says, "but it was a great generation, the first real generation of youth." It's when grungy meant “envious,” a holaholy was someone who objected to necking, and when things were good, they were hotsy-totsy.
But Goodman doesn’t attribute the “grungy = envious” factoid that he includes in his story to Dalzell, so we don’t know whether Dalzell endorses that equivalence or not. And in any event, we don’t have any hard evidence of when and where that usage has been recorded.
I don’t find it especially hard to believe that certain hipsters of the 1920s might have used grungy to mean “envious,” but I’d like to see some documentation that they did so before I accept the assertion as true. The unattributed, identically worded, skeletal coverage of the subject at other online sites does not inspire confidence in the rigor and authoritativeness of that coverage.
The newer meaning of grunge surely antedates 1965. In a Google Books search, I discovered that the term grunge-jumper appears (twice) in Jack Kerouac, The Dharma Bums (1958). Here’s the first instance:
Meanwhile, suddenly Japhy's sister Rhoda appeared on the scene with her fiancé. She was going to be married in Japhy's father's house in Mill Valley, big reception and all. Japhy and I were sitting around in the shack in a drowsy afternoon and suddenly she was in the door, slim and blond and pretty, with her well-dressed Chicago fiancé, a very handsome man. "Hoo!" yelled Japhy, jumping up and kissing her in a big passionate embrace, which she returned whole-heartedly. And the way they talked!
"Well, is your husband gonna be a good bang?"
"He damn well is, I picked him out real careful, ya grunge-jumper!"
"He'd better be or you'll have to call on me!"
A note in Gregory Stephenson, "Explanatory Notes to Jack Kerouac's The Dharma Bums*" in Polarity Magazine runs as follows:
grunge-jumper: the word grunge means any nasty substance or a dirty and distasteful person; whereas a jumper is a copulator. Hence, a grunge-jumper is one who is so indiscriminately or so desperately lascivious as to copulate with persons who are unclean or otherwise repellent.
Many instances of grunge and grungy in the sense of something grimy, grubby, dingy, and/or sludgy appear in a Google Books search for matches from the middle to late 1960s and the early 1970s, as you’d expect.
Also of potential interest is this very different sense of grunge cited in John Jones Design Methods (1970) [combined snippets]:
[Robert W.] Mann [in "Engineering Specifications for a Man-Computer System"] (1963) describes the long periods of systematic thinking that are often necessary in designing as 'crank-turning or grunge' and the shorter imaginative periods as 'creative peaks'. The creative peaks of a traditional designer may well be the occasions when he resorts to 'back-of-an-envelope' sketching; his grunge consists mainly of the making of scale drawings and detailed calculations. We may suppose that the skill of knowing when to switch from one to the other is something that a traditional designer develops for himself.
And similarly, we have this snippet from University of Michigan, Engineering Summer Conferences, Applications of Computers to Automated Design (1965):
Much but by no means all, of the methodology and techniques involved in the analytical, lower level, longer duration "grunge" or "crank turning" is very properly relegated to the machine. ... If you will we want our computer-aided design system to change our fanciful "CRO display" of a design process from occasional decision peaks inserted into long periods of noisy "grunge" into a time sequence where the "grunge" period is reduced toward zero and the opportunities for new decisions and judgements based upon analysis occurs again and again with far more possible reiterations, and the consideration of far more alternatives of concept, configuration, parameter values, etc. than is now permitted by limitations of schedule and budget
Here grunge sounds as though it might have different root words—perhaps grind and plunge—from those that underlie the word as it is usually used today.
And finally, in a search of the Elephind newspaper database, I found an early instance of grungy in use in a university setting in a party announcement by "the men of Alpha Chi Rho fraternity in the [University Park, Pennsylvania] Daily Collegian (February 8, 1964):
The Crow swung out to Bellafonte/ to produce some "Milk" and Cheese—/ Brought it back to our abode,/ to welcome our Rushees./ If you should be a Freshman/ Woe to you, dear Friend—/ But don't fret too much, son,/ The world's not at an end.../ (we have sexy Pepsi for you)/ So put on some grungy clothes/ just about anything will go—/ pick up your date and swing on out/ to meet the jolly Crow.