@Jack79, he could do that. Right now the syntax and semantics of the adverb are such that the adverb modifies the evil thing by saying how or in what manner it is evil. (We need an adverb because “evil” is being used as an adjective.) The sentence is not about his feeling.
In order to speak about the effect it has on him while still applying to the evil thing, it would have to be turned around and reoriented so that the attribute of the evilness is expressed in terms of an effect on others. If it causes him to seethe, then it would have to mean something like “causing-to-seethe-ly” or “seethe-makingly.” We don’t have any such word. Remember that “seethe” is being used figuratively, with “boiling” being an analog for an emotion, but it has to work first on a literal level. What causes something to boil? Heat. So we’d need a word that means “heating to the boiling point,” and we don’t have such a word available to be turned into an adverb.
So what does express this kind of quality as an effect on others? Let’s start with the other end, with the emotion. Seething isn’t an emotion; it’s a metaphor for an emotion. What is the emotion? anger? fury? revulsion? loathing? Those would be the sources of your adverbs if you were adding the emotional component. You’d build it this way:
The evil of certain people and ideologies {infuriates | revolts | maddens} me.
[To pick one]
It has an infuriating effect on me.
They are infuriatingly evil.
If I were his editor, I would not let him say this. It adds no meaning, and it diminishes the effect of the statement by turning it into something subjective. It takes the focus off the subject and makes it about the speaker. But this is the way to structure the adverb so that it expresses a feeling attributed to the speaker and not to the thing he is talking about.
“Unsettlingly” would be analogous to that: it would say that the evil thing he is describing has an unsettling effect on him.
We do have to remember here that we are talking about evil, which is generally taken to be not a good thing, which harms and angers and unsettles (etc.) people and which by virtue of being evil usually does not need much added explanation. I would say that the writer has a bigger task to perform in justifying calling any people or ideology evil, and so he’s going to need to hold an objective stance if possible. (Or maybe all he does mean is that those unnamed people and their ideology make him seethe—that is their crime, and not being evil per se.)
In morphail’s OED example, “seethingly” is used to modify the person’s anger: the friend is angry or is showing anger in a seething way. This makes sense and is consistent with the meaning and metaphorical application of the word.