You won't find a lot of information about the word adverbiously
out there. It doesn't likely show up in any dictionaries you can lay your hands, and on for one simple reason: Charles Dickens made up the word.
However, one can deduce what Dickens meant by examining the text that surrounds it. Adverbiously appears in this description of what Jerry Cruncher heard, or at least what he understood, in the courtroom in Book the Second, Chapter 2:
Charles Darnay had yesterday pleaded Not Guilty to an indictment denouncing him (with infinite jingle and jangle) for that he was a false traitor to our serene, illustrious, excellent, and so forth, prince, our Lord the King, by reason of his having, on divers occasions, and by divers means and ways, assisted Lewis, the French King, in his wars against our said serene, illustrious, excellent, and so forth; that was to say, by coming and going, between the dominions of our said serene, illustrious, excellent, and so forth, and those of the said French Lewis, and wickedly, falsely, traitorously, and otherwise evil-adverbiously, revealing to the said French Lewis what forces our said serene, illustrious, excellent, and so forth, had in preparation to send to Canada and North America.
Notice the repeated use of the phrase "and so forth." Certainly the courts would not have simply glossed over the legalese and traditions of the court, but since none of that really held any interest to Jerry (or to the readers), this description rightfully skips over the more tedious and boring parts of the court's proclamations.
Likewise, Dickens's use of evil-adverbiously is also a shortcut in the description. A list of -ly adverbs precedes it: ". . . and wickedly, falsely, traitorously . . ." Certainly other adverbs of an evil nature had been applied, either by the prosecutors or by the public, but because Jerry was more concerned with the outcome of the trial than with the particulars, Dickens again glosses over specific words to get more quickly to the action.
Evil-adverbiously, then, is a bit of wordplay that stands in for other -ly adverbs of a sinister nature that aren't important to the storyline, to the readers, or to Jerry Cruncher.