Assuming you don't want action-focused games, though some of those have some of the best drama. Keep in mind that the death of drama in SOL games started fairly early, with Da Capo. Finding mostly SOL games with actual drama in them is pretty hard.
Semiramis no Tenbin- Adversarial relationship between the protagonist and Ami for much of their relationship. While he tends to be more supportive with the other heroines, there is still a lot of personal drama in each route with the exception of Sunao's, which is essentially a bad ending.
Akagoei (series)- Most of the paths in Akagoei have the protagonist at odds with the heroine to some degree, though it varies from main heroine to main heroine.
Hikoukigumo no Mukougawa- Basically Eiri's path is what you want, no to the others.
Moshimo Ashita ga Harenaraba- Nakige but the drama in the heroine paths is pretty intense at times, somewhat dated visuals though.
Mirai Nostalgia, Chrono Clock, Hapymaher- Purple Soft games.
Konata yori Kanata Made- Might be too dated for you, utsuge. However, massive unavoidable drama in all paths.
Ruitomo- Varying levels of personal drama in each path.
Non-Fantasy/Sci-fi Minori games- Most of them have what you want, but I hate Minori, so I'm not going to go into details.
Ojousama wa Gokigen Naname- Butler protagonist, psychotic older-sister type antagonist, old rich family, chaos, greed, and general financial mayhem and plotting. Might be too much drama for you.
Soshite Hatsukoi ga Imouto ni Naru- I honestly recommend this one for a number of reasons, not the least of which because all the heroines and the protagonist are pretty mentally scarred and thus drama is unavoidable.
Oni ga Kuru- Drama, incest, perversion, etc... the relationships are pretty messed up in this one.
Oni Uta- Same scenario designer (though not writer) as above, similarly messed up.
Kanojo-tachi no Ryuugi- Messed up drama mixed with vampire siblings almost from the beginning. Same scenario designer as above.
Love Destination- Similarly messy story, different team. Guy reaches the end of his lifespan alone and gets dropped back in his distant past when he was a student, whereupon he tries to create connections. The drama can be pretty extreme, depending on the heroine/s.