Girl discovers that she is or becomes a fairy and has an enemies-to-lovers relationship with a prince of fairies

Tithe: A Modern Faerie Tale by Holly Black partially matches your description. Kaye Fierch, the main character, isn't an orphan, but no father appears in the novel, and her mother "is more of a friend to Kaye than a parental figure".

... had an enemies-to-lovers type romance with the prince ...

Kaye rescues Roiben, a faerie knight (and, as it turns out, the designated successor of his queen), but finds out that he is the murderer of a faerie who was her childhood friend. The enmity caused by that and by the revenge Kaye takes changes into mutual attraction as the ways of Kaye and Roiben keep crossing and Roiben eventually is able to rescue Kaye from becoming the Tithe needed to bind the Solitary Fey to the Unseelie Court.

... who either discovered she was a fairy ...

At that time, Kay has learned that she isn't human, but a changeling.

There was a war of sorts and possibly a power struggle between the fae ..

As it turns out, making Kaye the Tithe was the culmination of a long-running scheme to set the Solitary Fey free -- which in turn is part of a wider plan.

I remember the fairies had wings and tails ...

While don't recall tails, Kaye (when not disguised as human) and other faeries indeed have wings.

the human world was a little dystopian, though I don't believe there to have been a specific time period.

The novel is set roughly in the time it was published (2002) or not too long before (a single parent touring with a rock band does not really match any time before 1960 or so). The rock nomad background of Kayes mother, the problems and conflicts in Kayes environment, and the apocalyptic scenes after the liberation of the Solitary Fey definitely provide a dystopian touch.

I read this years ago on my nook (around 2010-2012)

"Tithe" was published in 2002.

They definitely slept in hammocks in trees as fae.

Sadly, this isn't found in the book, as far as I recall.

It was a little smutty.

A kiss on the buttocks, seduction, descriptions of SM -- does that count?

This is likely only a partial match, but the Paranormalcy books by Kiersten White match the main details.

Evie's always thought of herself as a normal teenager, even though she works for the International Paranormal Containment Agency, her ex-boyfriend is a faerie, she’s falling for a shape-shifter, and she’s the only person who can see through paranormals' glamours.

But Evie's about to realize that she may very well be at the center of a dark faerie prophecy promising destruction to all paranormal creatures.

So much for normal.

Evie is, apparently, an orphan. She learn (in the second book, I think) that she is half fairy (her father). As noted in the summary above, Reth, a fairy, is her ex-boyfriend, and continues to be a romantic interest. The setting is not explicitly dystopian, but it's evident early on that, without the IPCA keeping them in check, humanity is largely helpless against the supernatural. Parallel, the IPCA exerts an iron grip on the supernatural, freely meting out death sentences to those who break their rules, even if it's just accidentally breaking glamour. And there's an uneasy situation where the IPCA has prominent fairies enslaved via their true name, where it's obvious that if the fairies break free, they will be able to greatly damage humanity.

What does not match, as I recall, is the smuttiness. It's pretty normal YA fare without some indication of fumbling sexual experimentation, but nothing explicit.

I doubt this will be it because it was quite mainstream at the time but Wicked Lovely by Melissa Marr (2007) has a similar story.

At the start of the book, protagonist Aislinn is an orphan and lives with her grandmother. She already has a connection to fairies (inherited from her grandmother) in that she is one of the few humans that can see them. Then events of the story set in motion:

Keenan is the Summer King, who has sought his queen for nine centuries. Without her, summer itself will perish. He is determined that Aislinn will become the Summer Queen at any cost — regardless of her plans or desires.

There definitely is an enemy-to-lovers progression between the two characters but it's complicated by the fact that both Aislinn and Keenan have feelings for other characters as well. In the end (spoiler) Aislinn becomes the Summer Queen so she does become a Fae.

I haven't read the book so I'm not sure about how well other elements of your description fit.

About the smuttiness, I think it was considered quite mature reading at the time for a YA book, but again I can't tell firsthand.