The negative of "already" is usually "not yet", as you were taught, but "I didn't want to do it yet" means I didn't want to do it now or earlier (but I might want to do it later) and your example has a different meaning.
The person saying "I already didn't want to do it anyway" means that they had already decided that they didn't want to do it at all.
However, "I already didn't want to do it anyway" sounds a bit awkward to me and using both "already" and "anyway" in this sentence sounds odd too, even though those two words don't mean exactly the same thing. "I didn't want to do it anyway" would mean much the same. If "I already didn't want to do it anyway" is only part of a sentence, though, then the way it is phrased might make more sense in the context of the sentence as a whole.