I'm Japanese and one of big music fan, have been listened over a half million Japanese songs.
Even with アニソン(Anison=Anime song) alone, over 50,000 songs are listed in my playlist.
If you search for that genre with "泣ける曲(nakeru kyoku)"
at Google, YouTube and more hits a ton of songs like below,http://ur0.work/Ogr5
but it is different from the Japanese feelings, it also depends on the Japanese comprehension ability,
so it may be that the simple lyrics are more like ボカロ(Vocaloid) I think.http://ur0.work/OgtE
Also, "泣ける曲" has two, one is "emotional songs (sympathize with the world view of lyrics)" that you want,
but other one is "great songs (play techniques or vocals)".
I think that the thing you want to ask is the first one,
but you should remember that even great songs make us cry.
Those who are amazing songs are more commonly called "神曲(kami kyoku)".
Anyway, if looking for 恋愛ソング(renai = love songs),
you should add "片思い/片想い(kata omoi)" or "失恋(shitsu ren)" to keywords be the best.
Speaking of other sad tune themes, there are also "death" by sick, accidents, and war also usual.
This song is one of my favorite "ありがとう by KOKIA" , but has rare background.https://youtu.be/j71v5HnSP64http://j-lyric.net/artist/a00af13/l00a78b.html
Background : https://en.wikipedia.org/wiki/2004_Ch%C5%ABetsu_earthquakehttps://en.wikipedia.org/wiki/2004_Derailment_of_Joetsu_Shinkansen
This song is not a lyric theme with the theme of this disaster occurring,
but since there are no more optimal songs, it gained a strong empathy from net people.
You already know "most Japanese like Shinkansen", so "he" saved many passenger's life.
You do not know whether studying "God" to stay in various things(as 神道 =Shinto) in Japan,https://en.wikipedia.org/wiki/Kamihttp://lovesjapan.blogspot.com/2013/04/yaoyorozu-no-kami-god-of-japan....
but this is also close to that feeling, although the phenomenon (sense)
that attachment to the thing we normally use is not limited only to Japan, right?
One of famous TV anime "Angel Beats! (in 2010)" is filled with sorrow with both content and songs.https://en.wikipedia.org/wiki/Angel_Beats
In this year first season "Violet Evergarden" cried many anime fans not only of Japan,
but OP/ED songs are good(I like both), however, it was not enough in the impact as lyrics.https://en.wikipedia.org/wiki/Violet_Evergarden
I also would like to recommend this Anime "シゴフミ(in 2008)" and ED song "Chain by Snow*".https://en.wikipedia.org/wiki/Shigofumi:_Letters_from_the_Departedhttp://www.kasi-time.com/item-28565.html
This famous anime "ひぐらしのなく頃に(from 2006)", the song "You (or, Dear you)" is also very famous,http://j-lyric.net/artist/a04dda5/l0224c1.htmlhttps://en.wikipedia.org/wiki/Higurashi_When_They_Cry
but the content is "suspense" with killers so I do not recommend it.
Also, this is rare that there are plenty of arrangements for songs as official releaced sound.https://www49.atwiki.jp/aniwotawiki/pages/3360.html
This famous anime new series "宇宙戦艦ヤマト2199 (in 2013)",https://en.wikipedia.org/wiki/Star_Blazers:_Space_Battleship_Yamato_21...
first ED "愛詞(ai-kotoba) by 中島美嘉(Nakashima, Mika)" may good.https://youtu.be/iExx1EIbxVw
(not full version)https://www.uta-net.com/song/145093/
The songwriting composition of this song is made
by "中島みゆき(Nakajima, Miyuki)" called "歌姫(utahime, like a diva)" of J-pops/falks.https://en.wikipedia.org/wiki/Miyuki_Nakajima
The original is a work long before you were born,
but I did not think that it would create such a famous song in the 21st century.
There is no Japanese of the title of this song, but it has a meaning as "合言葉(ai-kotoba = watchword)".
Unlike lyrics made by young people, there is "深み = depth (人生の重み = weight of life)",
but at your age you may not understand yet.
中島美嘉 is also one of the famous singer who sings plenty of songs of sorrow lyrics.https://en.wikipedia.org/wiki/Mika_Nakashima
Just listening to all the songs on the official channel may make all the schedule for this weekend useless.https://www.youtube.com/playlist?list=PL9UqCZitNKr0M5zaLexUaJ3o8gws0LT...
Oops, forgot about simpler and famous love song.
"奏（かなで) by 藤宮香織(charactor name)（CV : 雨宮天）", anime "一週間フレンズ。" ED.https://youtu.be/pwSvPDNG0uMhttp://j-lyric.net/artist/a001fed/l0074e4.htmlhttps://en.wikipedia.org/wiki/One_Week_Friends
Original is "スキマスイッチ" but male vocal, they are also famous group singer.https://youtu.be/J5Z7tIq7bco
Just I love female vocal version than original, makes me feel softer.
However, this song is a man 's point of view, you should listen to the original.
And also I would like to recommend OP(not sad melody) and story of this anime.http://j-lyric.net/artist/a057eff/l031653.html
The last one, "aiko" is also famous singersongwriter of songs of sorrow,
her songs are not related Anime, but a lot of famous TV drama series used.https://en.wikipedia.org/wiki/Aiko_(singer
Personally, I want to make me cry in a song that is strongly impressed than a sad song.https://youtu.be/wbaILDE7Dcohttps://youtu.be/BDF25ONbdD4https://youtu.be/4DE4TH5cAf0
This is an ED of anime, but it is the same as "愛詞", so you may not be understood delicate emotions yet.https://youtu.be/f-TxAUiwo18http://www.kasi-time.com/item-77715.html