A couple pieces come to mind: It's Me, Eddie, a roman a clef by Eduard Limonov (who was one of the most baffling post-Soviet political figures in Russia). Limonov was a Soviet dissident (sort of - it's very difficult to place him among other 'dissidents' of the 1970s; he's not exactly Brodsky or Solzhenitsyn) who lived in squalor in New York.

Limonov's novel is a wonderful, profane narrative that can give us some glimpses to the treatment of 1970s emigres, but it's very important to distinguish these immigrants from those immediately postwar or after the Revolution (in no small part because there were relatively few immigrants after the early 1950s until 1971). Limonov is ambivalent about America and the USSR: the US fails to live up to its ideals, but he of course defected from the Soviet Union. Many of the Russian characters in the novel try very hard to not be Soviet:

[For John]...the desire not to be Russian, the scorn for Russia, for its people and language, are also familiar...

Limonov draws out the problems of the land of opportunity, which is largely his source of income - $278 monthly from welfare.

[I was sitting with] Brooklyn Bridge bums, and burst out laughing. Shit, and this is civilization. Why aren’t they afraid of the gigantic waves that will some day rise up from the slums of Brooklyn and the Lower East Side and fucking submerge the little islets where the feast goes on in time of plague, where the sounds of hollow music flow, bunny asses flit around practically bare, and my Elena walks accessible to all? And no provincial one-story America can fucking save anybody, all shall be as New York wants, my great and flaming city…

and often returns to this motif equating Soviets and Americans.

Boredom, despondency, and in the last analysis a joyless death – ordinary Russian people reeked of it. Americans reek of it now. New York is rotting around the edges. Its clean blocks are much smaller in area than the already boundless sea of uninhabitable and semihabitable neighborhoods, terrible in their state of near-wartime destruction...

...and the aspirations he has, or had.

I too yearn sometimes...I would have a fabulous house with a lawn, with flowers, with a wealth of household appliances, a clean little smiling American wife, a freckled, jam-smeared son in football cleats…

But why dream – it’s futile...

Great is New York, long are its streets, homes and apartments has New York of every sort. Whom I shall meet, what lies ahead, none can guess..

Tears of agitation well up in my eyes, as always when I am agitated, and I no longer see Madison Avenue below. It dissolves and runs.

“Fuck you, cocksucking bastards,” I say, and wipe away tears with my fist. Perhaps I’m addressing these words to the buildings around me. I don’t know.

“Fuck you, cocksucking bastards! You can all go straight to hell!” I whisper.

At times Limonov compares the American tourists of New York to the uncultured Soviet citizens; he compares the FBI and CIA to the KGB, and mocks the CIA's support of 'modern art.'

America gets even with its artists by other means than Russia does. Russia is also wising up, however. An exhibit at the board of the Soviet Artists’ Union by some friends of mine, artists of the extreme left, is a case in point: Russian administrators are learning from their American colleagues the more modern, humane means of killing art, namely, if you want to kill an artist, buy him.

The first American woman - a Jewish German immigrant - he sleeps with is on July 4 1976, but the maids of the hotel refuse to clean his room that was only offered to him under the pretense he was Jewish. The Americans pressure him to adopt a conventional career, which is exactly why he left the Soviet Union.

Now, Limonov is a suspicious figure. He is involved in radical groups, is openly homosexual, revels in upsetting others, and so on - there's a reason the motto of his Russian political party was Da, smert'!. But the Americans in the novel don't adopt a particularly opposition to the idea of Soviet emigration. Andrei Sakharov won the Nobel Peace Prize in 1975 and Solzhenitsyn won the Literature Prize in 1970.

However, the Congress of Russian Americans was founded in the 1970s to distinguish between Soviet and Russian, arguing that Russians should be seen as victims of the revolution and not the progenitors of communism. There certainly was anti-Russian and anti-Soviet discrimination, but there simply weren't a great number of immigrants in the 1960s. It was exceptionally hard to leave the Soviet Union either legally or illegally. In 1970, in fact, more than a dozen refuseniks (a group often accused, baselessly, of tuneyadstvo - the way of life that Limonov adopted) attempted a plane hijacking in order to leave the USSR; this was a very public and tremendously embarassing event for the Soviet Union that forced it to increase avenues to legal emigration, particularly of Jews.

By 1971 Soviet Jews were allowed to freely emigate, and over 100,000 ended up in the United States. There's a joke from the 1970s to this effect: Yevreyskaya zhena - eto ne roskosh', a sredstvo peredvizheniya, a Jewish wife is not a luxury but a way to leave". So while Russians emigrated in this manner - like Limonov - they were allied to Jewish immigration and joined, oftentimes, Jewish communities. Unfortunately I can't speak particularly well to the reception of Jewish immigrants in the 1970s.

Brodsky's Nobel Speech is another look at the reception of a very different Soviet emigre. He speaks about the place of literature in the Soviet Union:

If art teaches anything (to the artist, in the first place), it is the privateness of the human condition....It is for this reason that art in general, literature especially, and poetry in particular, is not exactly favored by the champions of the common good, masters of the masses, heralds of historical necessity.

and less obliquely of the Soviet Union writ large

Although for a man whose mother tongue is Russian to speak about political evil is as natural as digestion.... “How can one write music after Auschwitz?” inquired Adorno; and one familiar with Russian history can repeat the same question by merely changing the name of the camp – and repeat it perhaps with even greater justification, since the number of people who perished in Stalin’s camps far surpasses the number of German prisoncamp victims. “And how can you eat lunch?” the American poet Mark Strand once retorted. In any case, the generation to which I belong has proven capable of writing that music.

Brodsky's poems, however, aren't particularly political. He was fostered by academics in New York and Ann Arbor. Brodsky, Solzhenitsyn, Limonov - and even figures like Stalin's daughter, Lana Peters - aren't typical immigrants applying for jobs like you might imagine. There just weren't many 'typical' immigrants from Russia at the time.

For experiences closer to the average Russian, we might turn to those who fled immediately after WWII, or especially those who fled and were expelled between 1917 and 1925. After WWI, the First Red Scare is probably closest to what you're imagining, while after WWII Russians represented a minority of European refugees to America. I imagine it is these groups that turn into the Congress in the 1970s, so there is still anti-Russian sentiment then.