Yes, even if you know for a fact that you'll reject both of them. They're too important for everything else that happens in continental philosophy. Almost everyone else is responding to them or interpreting them in different ways.

For Heidegger: you can get away with only reading the first division of Being and Time but I'd also prioritize "The Origin of the Work of Art" and "The Question Concerning Technology," among other pieces included in Heidegger's Basic Writings. Even if your main question as a reader is "What's fashy about this ostensibly non-political text?" it's important. I'm not going to claim it's necessary for everyone to read him, but since you're interested in philosophy, you're already researching existentialism, and you presumably have mountains of leftist books on your reading list that have been influenced, implicitly or explicitly, by Heidegger's thought - yes, you should read him.

Re: Husserl: I know a few people who've read him and I don't know if any of them recommend the experience. The folk I know who are into phenomenology are much more fond of Merleau-Ponty.

For Nietzsche: Genealogy of Morals is his most straightforward and necessary text. The Birth of Tragedy is great; The Gay Science and Beyond Good and Evil are at least fun to flip through; Twilight of the Idols is an underrated starting point. There are good reasons to loathe Nietzsche and his influence but he's an absolute all-timer as a writer and aphorist, and if he says something deeply horrible he probably has something that says the exact opposite elsewhere in his corpus. There are lots of ways to read him. For example, he's obviously a misogynist but there are books out there that offer feminist interpretations of him. Is this ridiculous? Maybe. But there's a reason he inspires such contortionist versatility in his readers. He contains multitudes.

For Cioran: my favorite is a A Short History of Decay. His first book, written in Romanian, On the Heights of Despair, is frequently excellent. If you've read even a bit of Nietzsche you'll see where Cioran gets it from. I don't know where else to go for nihilist thinkers, but Eugene Thacker, who does some Cioran introductions, has a book on pessimism (Infinite Resignation) that might direct you to others.

I found this Red Sails piece on Nietzsche interesting, also mentions Heidegger

https://redsails.org/really-existing-fascism/

No and yes.

Edit: philosophizing with a hammer and sickel might be a good companion to nietsche.

Someone poasted this not long ago.

https://mronline.org/2024/03/29/why-the-left-should-reject-heideggers-thought-part-1-the-question-of-being/

you can read losurdos 1000 page behemouth study of nietzche: "nietzsche the aristocratic rebel"

@Wertheimer@hexbear.net is correct about the correct readings from both Heidegger and Nietzsche. Even if we should reject a Nietzschean will to power, recognizing it is important.

The end of the (I think second?) aphorism of Geneaology of morals is an all time mic drop.

This self-overcoming of justice: one knows the beautiful name it has given itself – mercy; it goes without saying that mercy remains the privilege of the most powerful man, or better, his – beyond the law.

Reading a synopsis for Nietzche is likely for the best if you just want the general vibes. His writing style is really solid, mind you. Also Sprach Zarathustra is poetic, even if it's up its own ass threes times over and I wanna say it was Beyond Good and Evil that discussed Dionysis as an avatar of primordial chaos. Short read, that latter one.

I’m in the same place basically. I think I want to actually read the source material at some point but I recommend (though haven’t read) Losurdo and Lukacs. Btw, I don’t know about any real nihilism, as Nietzsche isn’t one. Camus is kind of like nihilism but cool and constructive. Nietzsche hated nihilism and thought it’s bad that modernity killed [belief in] god. He thinks we gotta return to pre-judeo-Christian morality where the strong (apparently him) rule the weak.