- "I always do my homework." Does the adverb always modify do or do my homework?
- "I went to the movies quietly." Does the adverb quietly modify went or went to the movies?
Well, you already know that the adverb modifies the verb. Your question is whether do-~'s-homework and go-to-the-movies are elaborate verbs?
Well, yes and no.
If you were going to algorithmically translate from English to other languages, you would need to treat do-~'s-homework as a single verb separate from do because not every language phrases this the way English does. For example, in Chinese, students "write" their "obligatory work" (写作业) and you'd need to understand that all as a phrase and not try to do it piecemeal.
Within English, no, do and go are the verbs, the others aren't, and you already know that.
The thing that's confusing you is the difference between the grammatical and the semantic content. Semantically, yes, the phrases describe two actions that are separate from do and go but, owing to English vocabulary, those actions are expressed as verbs modified by their object or a prepositional phrase.
Put simply, there are no other choices.
Adverbs can modify a whole host of grammar constructs. In either example you gave the verb is the only thing they could modify. Not the entire phrase or sentence.