I just noticed that a meaning of a sentence can change depending on
where an adverb is placed. I'm not sure if there is a rule forbidding,
or if there are even examples for that.

See this:
   {not bIQong 'e' vISov.}
   "I know that you never sleep"
vs.
   {bIQong not 'e' vISov.}
   "I never knew that you sleep."

Can I do that?

TKD 6.7 (add) says that "the adverbial precedes the object-verb-noun
construction". In the above example, {'e'} is the object of the verb
{Sov}, so I think it's correct. – Or did I forget something?

I somehow feel there is a canon example for this (x not 'e'), but I just
can't remember.

There is, but we didn't get it until paq'batlh.

We had long ago worked out that the 'e' reH SkyBox example was problematical. We simply reasoned it out: if 'e' is always the object, and if adverbials come before the object, then when modifying a verb whose object is 'e', the adverbial should come before the 'e'. And since 'e' is only ever used in sentence-as-object constructions, that means that the correct form, at least according to all the rules we've been given, is S1 A 'e' V2, where S1 is the first sentence, A is the adverbial, and V2 is the verb of the second sentence.

Then we got the SkyBox example, and that didn't conform to our idea. But that sentence has other grammatical errors as well. I believe that when Okrand wrote the SkyBox example, this was when he was still not very practiced at translating into Klingon, and he was making a common newbie mistake: he was thinking of 'e' as a conjunction between two sentences, must the way that that is the corresponding conjunction in English. He was imagining S1 'e' S2. And if you imagine 'e' as a conjunction, naturally you're going to imagine the adverbial of S2 is part of S2.

We also had reH DIvI' Duj vISuv vIneH from Star Trek V, and this is even more problematical, since it looks like it's saying I wanted I fight a Federation ship forever: that is, the forever is modifying the fighting, not the wanting. Either that, or the adverbial is being applied to the sentence-as-construction as a whole: A (S1 S2). And of course, it's a neH sentence-as-object construction: without an 'e', does that change the rules? Are they special with regard to adverbials?

So we had no clear canonical example, and we only had the logic of the rules. Then we got paq'batlh, and I think that's when we got our first unambiguous examples.

SuvwI' DameH puqloDwI'
    vIghojHa'moH DaH 'e' vItlhoj
    bIQ lungaS 'aDDu'Daj

I see now, I have failed
    To raise my son a man.
    Water flows through his veins.

The DaH is definitely meant to modify tlhoj.

We also have examples in paq'batlh of adverbials that are definitely meant to modify the first verb:

tugh Hegh 'e' Sov moratlh
Morath felt the end was near.

The tugh is certainly meant to modify Hegh.

So we do have canonical examples of S1 A 'e' V2, but we didn't have them until much later than you were thinking.

-- 
SuStel
http://trimboli.name