Learn English – What’s the origin of “crapweasel”

Your question is conflating two different meanings of "way".

"By the way" is literally "by the side of the road", but the OED cites its figurative use ("Incidentally, in passing, as a side-topic.") from 1556. I don't believe a single authoritative source will ever be identified for it, as I guess it was in common speech before 1556.

"Sidetrack" uses the same metaphor, as you imply.

But "roundabout way of speaking" is "roundabout [way of speaking]", and uses "way" in the sense of "Manner in which something is done or takes place; method of performing an action or operation." (the OED's sense 14 a).

The Oxford English Dictionary doesn't have doublespeak as a separate entry; I can only make an educated guess. The word double-talk already existed in 1938, according to the OED, and was originally American. But apparently it meant simply "deliberate gibberish" then, i.e. not merely ambiguous language, but nonsense talk that listeners were supposed to know meant nothing, uttered to comical or artistic effect.

1938 New York Panorama (Amer. Guide Ser.) vi. 156 Of late a humorously conceived system of language corruption called double talk ... has made itself felt. Ibid. 157 Double talk is created by mixing plausible-sounding gibberish into ordinary conversation, the speaker keeping a straight face or dead pan and enunciating casually or off the cuff.

1941 Time 16 June 61/1 Thirteen recorded versions of this pandemic double-talk ballad are available.

1945 H. I. Phillips Private Purkey's Private Peace xxii. 129 We got the right slant on bullies, greaseballs, double talkers, supermen, and dopes.

1948 Auden Age of Anxiety vi. 125 And all species of space respond in our own Contradictory dialect, the double talk Of ambiguous bodies.

After Orwell's famous novel 1984, published in 1948/1949, double-talk apparently acquired the (additional?) meaning "deceiving language", i.e. language that is deliberately ambiguous, or language that appears to mean one thing but in fact means another.

1950 Amer. Speech XXV. 190 Back in Tsarist times Lenin and his associates inaugurated this double-talk and double-writing—especially the latter—in order to deceive the Tsarist censors and police.

1952 C. Day Lewis tr. Virgil's Aeneid ix. 203 You'll find no Atridae here, no double-talking Ulysses.

This is probably based on Orwell's newspeak and doublethink. His novel was (partly) a critique of the practice and propaganda of the totalitarian regimes of his time, Fascist Germany and Stalinist Russia.

Newspeak is new language invented by politicians as propaganda, in order to influence people's thinking by changing their language (one of the principal mechanisms studied by Postmodernist philosophers and sociologists). The word is invented by politicians in the novel and is an instance of itself. By drastically reducing vocabulary, and making new, simpler words from the remainder, they hoped to control the people more effectively. Many modern euphemisms could be called, and are often called, newspeak, like "differently abled" and "Secretary of Defence".

1949 ‘G. Orwell’ Nineteen Eighty-Four i. 51 Syme was a philologist, a specialist in Newspeak. Indeed, he was one of the enormous team of experts now engaged in compiling the Eleventh Edition of the Newspeak Dictionary. Ibid. ii. 133 Do you know the Newspeak word goodthinkful? Ibid. App. 299 Newspeak was the official language of Oceania and had been devised to meet the ideological needs of Ingsoc, or English Socialism. In the year 1984 there was not as yet anyone who used Newspeak as his sole means of communication, either in speech or writing.

Doublethink is somewhat related to what modern psychology calls cognitive dissonance: first accepting one fact as true, then another, contradictory fact—without critical self-assessment, which would ordinarily result in elimination of one fact or the other. Orwell's politicians try to effect doublethink by means of using and propagating ambiguous language.

1949 ‘G. Orwell’ Nineteen Eighty-Four i. iii. 37 His mind slid away into the labyrinthine world of doublethink. To know and not to know, to be conscious of complete truthfulness while telling carefully constructed lies, to hold simultaneously two opinions which cancelled out, knowing them to be contradictory and believing in both of them, to use logic against logic, to repudiate morality while laying claim to it, to believe that democracy was impossible and that the Party was the guardian of democracy.

Considering the change in meaning of double-talk soon after the publication of Orwell's novel, and the emergence of doublespeak to mean the same a few years later, your sources (reasonably) assume that the word was deliberately coined as a (semi-)portmanteau of Orwell's newspeak and doublethink. The OED has 1957 as its earliest quotation:

1957 ‘M. Buttle’ Sweeniad ii. 55 In the literary weeklies, the languages of criticism and theology have become one and book reviews all sound like sermons written in the most holy ‘*Double-Speak’.

1961 W. Kaufmann in G. E. Myers Self, Relig. & Metaphysics 99 The theologians have a way of redefining terms in rather odd ways, and frequently engage in something best called double-speak: their utterances are designed to communicate contradictory views to different listeners and readers.

The asterisk and quotation marks in the first quote might indicate that the word was then quite new and explained in a footnote.