It seems as if you are confusing some things. You have to clearly distinguish between adverbs and adverbials.
Adverbs usually modify adjectives, verbs and other adverbs. Adverbials (in your example you used the term adverb clause = a clause whose head (main word) is an adverb), however, can have a variety of other functions.
There are for example adverbials of time, place, concession; final adverbials, adverbials of manner, to name just a few.
Now let's turn to your sample sentences/phrases:
"I am so hungry that I can finish the whole food."
adverbial of cause
"He appeared for the test though he was ill."
adverbial of concession
"She was as pretty as a rose."
(part of comparison)
Adverbials are the optional elements in the sentences that provide background information on when, where, why, and how the event described by the verb and its objects takes place.