The 1841 British edition accidentally replaced the past participle drunken with the alternative shortened past participial form drunk. This reading was not continued in the following LDS edition (1849).
The past participial form drunken rather than drunk is consistently found in adjectival and passive contexts in the Book of Mormon text—that is, when the helping or main verb is be (or is at least implied). There are 17 examples of this usage in the text, all of which refer either directly or indirectly to alcoholic intoxication, as in Mosiah 22:7 (“when they are drunken and asleep”) or Ether 15:22 (“they were drunken with anger even as a man which is drunken with wine”). Thus the use of drunken rather than drunk is correct here in 2 Nephi 27:1.
On the other hand, there are six cases of the verb drink in perfect contexts (that is, when the helping verb is have). For these cases, we get more variation in the Book of Mormon text. In one case, an Isaiah quotation, we have both drunk and drunken:
Note here that Oliver Cowdery, when he copied this Isaiah passage, initially wrote “hast drunk” instead of “hast drunken” for the second occurrence, perhaps because he had just written “hast drunk” for the first occurrence. Oliver later corrected drunk to drunken, perhaps when he proofed 𝓟 against 𝓞 (the level of ink flow is slightly heavier). The original manuscript is not extant here but probably read the same as the corresponding King James text (“hast drunk ... hast drunken”).
Elsewhere in the text, we have either drunk or drank (but not drunken) as the past participial form in the perfect:
In the two original cases of “have drank” in 3 Nephi, the simple past-tense form drank was extended to the past participial form. Such usage is common in the Book of Mormon text, as in “we had came down” rather than “we had come down” (see 1 Nephi 5:1 for discussion of this example). For complete discussion of this usage, see past participle in volume 3.
Summary: Maintain the use of drunken rather than drunk whenever the text uses the past participial form of drink to refer to states of drunkenness (here in 2 Nephi 27:1 and in 16 other places in the text).