1) Two examples of American mModernist writers are John Steinbeck and Vladimir Nabokov.

In this regard, tThis essay will contrast these two novelists and their foremost works: The Grapes of Wrath and Lolita.

This essay will compare their contexts and influences, always framed underby the concept of Modernist Literature.

2) These counter-cultural writing styles further demonstrate at what levelhow the disintegrated state of society affected the work of writers at thate time.

Your sentence makes sense but it is very wordy

In this regardBecause of this, writers with a voice emerged to advocate for those without power.

I don't think "In this regard" is often used in English

In this category of writers bold enough to denounce social issues, John Steinbeck arises as one of the most important voices in the America of theof early 20th century America.

A writer whose books tapped into socio-political issues, relevant forto his time.

3) On the whole, both writers, the same as and their books have transcended time to become everlasting classics.

Using two commas here looks like a nonrestrictive clause. With a nonrestrictive clause you should be able to remove the clause from the sentence. Eg On the whole, despite the efforts of people to ban them, these books have transcended time.

On the other hand, Nabokov, has transformed into an icon of contra-cultural literature, not only for the subjects that he covered but also for the innovative resources he included in his books.

Both different in style but similar in essence, since both have become a referents in literature and synonyms of mastery.

2) These counter-cultural writing styles further demonstrate at whatthe level theof disintegrated state of society that affected the work of writers at that time.

In this category of writers bold enough to denounce social issues, John Steinbeck arises as one of the most important voices in theof America ofin the early 20th century.

3) On the whole, both writers, the same asalong with their books have transcended time to become everlasting classics.

On the one hand, Steinbeck has come to be an icon of rebellion against social injustices, ideas that he imprinted throughout all his work.

On the other hand, Nabokov, has transformed into an icon of countraer-cultural literature, not only for the subjects that he covered but also for the innovative resources he included in his books.

Both are different in style but similar in essence, since both have become a referent in literature and synonyms of mastery.