TLDR: It does not matter what part of speech which you or anybody else calls how; it only matters what it does. How is a wh-question word, which are in a category of their own. They can be called into service to fulfil many different gramatical roles. Asking after the condition of a noun demands some sort of noun modifier (or actual substantive) in response, which is typically an adjective — just like here.
The way to determine the part of speech played by any word in a phrase or sentence is not by looking it up in a dictionary, but rather by examining the job which that word is doing in the actual sample being scrutinized.
So for example, we have all these different uses of how:
- I don’t care about the hows or the whys; I just care about the results.
- How so sir, did she change her determination?
(Shakespeare, 1592, Merry Wives of Windsor; ɪɪɪ. v. 69)
- And so drew Argo up, with hale and how, On the grass.
(William Morris, 1867, Jason and the Argonauts, x. 587)
- I couldn’t figure out how to open the contraption for the life of me.
- Bob Cratchit told them how he had a situation in his eye for Master Peter.
(Charles Dickens, 1884, A Christmas Carol, iii)
- Do you know any good how-to books on crochet?
- Coming upon the Lone Ranger with his back turned, Tonto startled him with the traditional Sioux greeting, “How, Kemosabe.”
- What need ye hech and how, ladies? What need ye how for me?
(Mary Hamilton, 1889, in Child Ballads)
- To the top of Great How did it please them to climb.
(William Wordsworth, 1800, Rural Architecture, 4)
- Her acceptance of your apology depends on how you present it.
- How silly!
- How the hell did you ever do that?
- How about a picnic in the park today?
- How did the story turn out?
- How many sugar packets would you like?
- I put my whole back into it, and how!
- How now, Malvolio?
(Shakespeare, 1601, Twelfth Night, ɪɪɪ. iv. 16)
- ‘Will you join us in a little conspiracy?’
‘How do you mean conspiracy, young man?’
(Thackeray, 1849, Pendennis, lxxiv)
- How is little Johnny coming along in his studies?
- How does that that song go?
- How do you feel?
- How are those tomatoes?
- How was show last night?
(Several examples omitted for brevity.)
As you see above, even when used as a wh-question word at the start of a question, how can fulfil many different roles.
When an interrogative sentence of the inverted form Wh-V-S? gets flipped around into normal word order and the wh-word replaced by something discrete, so S-V-X, that X is not always an adverb or adverbial phrase. It can be an adjective:
- I feel strong and healthy.
- These tomatoes are tasty.
- Last night’s show was boring.
When that happens, it is because the original how was not asking an adverbial question like “by what means”.
Rather, it was asking about the condition or state of the subject. That’s why those answers are all adjectives: the verbs are copulae and so the predicate complement is adjectival not adverbial.
If this makes you classify how as an “adjective” here, then so be it. It does not matter.
You really should not get so hung up on parts of speech, particularly those from that or this dictionary. They are nothing but arbitrary assignments that vary widely according to the purpose of whoever is making them. They serve little real purpose — and here, I believe, none at all.
It does not matter what you or anybody else calls how; it only matters what it does.