The sentence that has an adverb phrase that modifies a verb is "in Mammoth Cave National Park you can tour the caves". Prepositional phrases can function as an adjective, an adverb or a noun. When it functions as an adverb it is telling us something about the relationship between the object of preposition head of the PP(prepositional phrase) and the verb.
In order to be sure if the PP is functioning as an adverb phrase, you should look at the object of the preposition, find which other word in the sentence is related to this noun or pronoun. If the other word is the verb, then that prepositional phrase is beheaving as an adverbial phrase.
Adverbial phrases answer the following questions as regards the verb: when, where, why, how, to what extent and under what conditions.
In Mammoth Cave National Park you can tour the caves
The PP is giving information about the place, the where.
Nevertheless, I would not say that it is an adverbial phrase but that it is a prepositional phrase acting as an adverbial phrase.