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?