Anywhere is a fused head of a noun phrase here, similar to anyplace. This categorization follows its use as object in constructions like
Have you got anywhere to spend the night?
As you've noticed, it takes post-head dependents as you would expect in other noun phrases headed by compound determinatives like somebody special or something new.
It can, in addition, take a relative clause, something that would be impossible were it an adverb.
You know I don't eat anywhere that requires a tetanus shot or a
Further, it can be the object of a preposition, something adverbs categorically shouldn't be able to do (with some small exceptions like until recently).
Money is power in Thailand - why should Thailand be different from
Go as a verb of motion requires a locative complement, usually in the form of a preposition phrase, but allows compound determinatives like anywhere, nowhere, somewhere as well. It does not, however, allow what are clearly adverbs (ending in -ly) as complement.
*They went internationally.
Yes, most dictionaries classify it as an adverb, but there is little reason to do so other than the fact that it can function as an adjunct (your adverbial) in clause structure. Word classes should be assigned based on the internal structure of the phrase the word in question heads, not functions the phrase it heads can take on in clause structure.