Moon Lovers: Scarlet Heart Ryeo (2016)

Both are about a modern day FL who travels into the past and finds herself unexpectently falling in love.

The beautiful female lead and the young king were deeply in love.
Due to stupid young king and palace conspiracy, dirty politician, they were apart
Can’t get my eyes off the queen
Beautiful historical drama

Both are about a modern day FL who travels into the past into the body and identity of another girl.

Both were historical drama.
Both female leads were beautiful.
The main lead were deeply in love
However, ending were sad

I recommend this both drama because , Male lead and Female lead
try to solve problem
In one drama explosion and another student murder/suicide

When a total eclipse of the sun takes place, Go Ha Jin is transported back in time to the start of the Goryeo Dynasty of Korea during King Taejo's rule. She wakes up in the body of the 16-year-old Hae Soo and finds herself living in the house of the 8th Prince Wang Wook, who is married to Hae Soo's cousin. She soon befriends several of the princes and meets the ostracized 4th Prince, Wang So. Although knowing she should not get involved in palace intrigues over the succession to the throne, she inadvertently becomes a pawn in the struggle, as several of the Princes fall in love with her.

Both are about a modern day FL who travels into the past and finds herself in involved with at least one prince. Both have the same sad ending.

Actually it is not a historical period drama like this one yet you can watch it if you hate the end like me. Indeed, Scarlet Heart 2 is the second part of Scarlet Heart chinese version. In this version , you'll see the two main leads in the 21st century.

Both are about the FL traveling to the past and unexpectently falling in love with the ML.
........................................

Both are about a modern day FL who travels into the past and finds herself in involved with at least one prince.

More then anything, because of historical facts. if you got interested in historical events from Scarlet Heart, you should give it a try ( like I did). This drama takes time just after Gwangjong's reign, and starts with story of Gwangjong's son, Gyungjong. There're some more couple of characters that you will recognize from Scarlet Heart.. As well, all the tragedies start because of the battle for the throne...

Both drama have:
Tragedy, palace intrigues, very angsty and complicated romances, betrayal, loss yet at point they are some very lighthearted moments. Also heavy on the multiple love storylines.

In the late Joseon dynasty, counterfeits were circulating around. Detective Ahn of Left Security Station and a passionate newcomer, Nam Sun, try to trace the source of the counterfeiting. They chase the Secretary of National Security and his man 'Sad Eye' as suspects. A predestined battle between the best woman detective in Joseon, Nam Sun, and a mysterious assassin 'Sad Eye' is unfolded. A magnificent, chivalric film that showcases Lee's style

Quiet and gentle, Min Jae Yi is everything a good daughter of a prominent family should be. Engaged to the son of the Second State Councillor, Jae Yi knows that the only way to bring herself and her family honor is to accept her fate, which she does with grace and charm. Devoting herself to learning everything she needs to know to become a good wife and mother, Jae Yi dutifully prepares for the future. Sadly, destiny has other plans for her future. Four days before her wedding, Jae Yi’s family is murdered. With no one else to blame for the unexpected tragedy, Jae Yi soon finds herself the object of baseless and horrifying accusations. Blamed for the death of her family, Jae Yi might not have had much of a future at all, if fate hadn’t decided to intervene. Bringing the lonely crown prince, Lee Hwan into her life at just the right moment, Jae Yi is saved from a dismal fate, but her salvation comes at a price. Plagued by a mysterious curse, Lee Hwan agrees to help Jae Yi, but only if she agrees to help him break his curse in return. Unable to refuse such an offer, Jae Yi agrees to the prince’s terms, setting them both on an unexpected path toward love. But what chance does their love have of surviving when one of them is a former murder suspect and the other is a cursed prince?

Both are about a person from a different era time traveling. One involves past to future, the other future to past. Both have sort of an open or bittersweet ending.