Grammar – Use of ‘Each’ and ‘Every’

But every time there's a thunderstorm around Marshfield, they say you can hear his rolling voice in the hollows of the sky.

Let's simplify the sentence to make it easier to analyse:

Every time there's a thunderstorm, you can hear his voice.

The phrase every time there's a thunderstorm is a temporal Adjunct (of frequency). Some grammars call Adjuncts "Adverbials". Notice that you can move this phrase around:

• They say you can hear his voice [every time there's a thunderstorm.]

In terms of what kind of phrase this is, it's a noun phrase. The head noun is the word time. This noun has a Determiner, the word every. Determiners are words like the, a, my, this and so forth. The word time is also being modified by a restrictive relative clause there's a thunderstorm. We could rephrase the sentence like this:

• Every time [that there's a thunderstorm], you can hear his voice.
• Every time [when there's a thunderstorm], you can hear his voice.

The word there is the Subject of the clause there's a thunderstorm. It is analysed as a pronoun in modern grammars such as The Cambridge Grammar of the English Language (Huddleston & Pullum, 2002). There are many reasons why. One reason is that it is repeated in question tags at the end of sentences:

• There are many thunderstorms, aren't there?

This is a different word from the locative word there that we see in sentences like I'll meet you there.

It seems then that the Original Poster is correct (apart from the small detail that every time ... is an Adverbial, but not an adverb - an adverb is always a single word).

Hmm… not even grammarians agree which part of speech is my in “my book”

• Possessive determiners constitute a sub-class of determiners which modify a noun by attributing possession (or other sense of belonging) to someone or something.

• They are also known as possessive adjectives, although the latter term is sometimes used with a wider meaning.

• Examples in English include possessive forms of the personal pronouns, namely: my, your, his, her, its, our and their, but excluding those forms such as mine, yours, 'ours, and theirs that are used as possessive pronouns but not as determiners.

• The words my, your, etc. are sometimes classified, along with mine, yours etc., as possessive pronouns or genitive pronouns, since they are the possessive (or genitive) forms of the ordinary personal pronouns I, you etc.

• However, unlike most other pronouns, they do not behave grammatically as stand-alone nouns, but instead qualify another noun – as in my book (contrasted with that's mine, for example, where mine substitutes for a complete noun phrase such as my book).

• For this reason, other authors restrict the term "possessive pronoun" to the group of words mine, yours etc. that substitute directly for a noun or noun phrase.

• Some authors who classify both sets of words as "possessive pronouns" or "genitive pronouns" apply the terms dependent/independent or weak/strong to refer, respectively, to my, your, etc. and mine, yours, etc.

• For example, under this scheme, my is termed a dependent possessive pronoun and mine an independent possessive pronoun.

Wikipedia