grammar - How do adverbs denoting amounts such as たくさん and ちょっと work?

Japanese counters such as 一つ are adverbs even though they describe nouns such as in the following sentence.

りんごを一つ食べた

It is clear that 一つ refers to the amount of apples. However, what do adverbs such as たくさん and ちょっと refer to when they are used, the amount of nouns the verb was done to or the amount of the verb that was done? For example

りんごをたくさん食べた

Does たくさん refer to the amount of apples eaten or does it refer to the amount of the verb done such as how "a lot" describes how much running was done in a sentence like "yesterday, I ran a lot"? Some other examples are

人がたくさん死んだ

Many people died.

私はたくさん死んだ

I died a lot.

I am not sure if these interpretations are correct, mainly because I don't know if たくさん as an adverb describes the noun or the verb. The former has たくさん seemingly describe the amount of nouns, but the latter has たくさん describe the amount of the verb done. ちょっと is also confusing. りんごをちょっと食べた sounds natural, but 人がちょっと死んだ doesn't sounds natural and does not feel like an opposite of 人がたくさん死んだ.