If you haven't seen it yet, watch Sakamichi no Apollon. The anime I ALWAYS recommend to anyone. The anime I would watch with my family if they asked "so what is this anime thing u always watch?". The anime you literally cannot dislike. It's so good.
Set in the 60s, it's about a secluded boy. Coming of age story but... just watch it. It has a lot of Jazz elements and if you're not hooked to Moanin' by Bill Evans by the end, well slap me silly and call me Nancy.
I would recommend Barakamon if you're looking for something ongoing. It is entirely my type of anime and I think you would enjoy it too. It's about this city guy who goes out to the country to better his art (himself). (8/10)
I think Space Dandy is one of the top3 comedys I've ever seen. By the creators of Cowboy Bebop. Really funny and artistically interesting. Very oldschool, not quite emotional in the sense you're looking for but hey laughing is good ya kno + it has its moments. (9/10, both seasons)
Watch Tsuritama. It's a drama and it's so well done. It's one of my favourites of all time and in my humble opinion (almost) a masterpiece. It's just really, really, really good on all fronts, and I think you would like it. It can also be very anime-esque in its humor and exaggeration, but if you weren't looking for that, then just watch TV shows ya kno? (10/10, but I jump it to 9 sometimes)
Nagi no Asukara is godlike... it's one of the best scriptwriting I've seen in anime (best?), and it's very modern. The artstyle is repulsive in that faces are uguu moeblobs, but outside of that there is sceneryporn by the million, so it's not that bad, honestly. Very character driven show. (10/10, maybe 9)
There's Spice and Wolf, which is a romance story wrapped in a medieval setting focused on economics. I would so so so so so SO recommend picking up the novels for this, but watching the anime first is not a bad thing whatsoever. It's really well done (even though I think it gets
absolutely blown out of the water by the novels) and one of the best romance anime I've ever seen. (10/10, both seasons)
Chihayafuru is something I have to mention. Technically a sports anime, but as someone who's competed in Smash I think that aspect is VERY VERY VERY relatable, at least it was to me. The tension of going to a tournament, losing to specific gimmicks and discovering the trick too late, or winning because of all your practice in the matchup. That stuff. The real focus though is the character relations, love, and all that little girl crap. I absolutely adore it, though. (10/10, both seasons)
Another one I recommend you check out is Natsume Yuujinchou. This one actually, in my opinion, is similar to Tsuritama but takes an episodic approach. It's like every episode (or sometimes two) is this little story with its own morals. You can see how everything affects the main character and how the world grows. It used to be my favourite anime series of all time, and still possibly is. (10/10)
If you liked Natsume, watch Hotarubi no Mori e, it's a 45 minute short movie that's incredibly excellent, same creators. (10/10)
Also, this one might fit you - Isshuukan Friends (translated: One Week Friends). It's about this girl who forgets all her friends at the start of a week, and a guy trying to stay friends with her. It's truly bittersweet, cute as ****, and can be an emotional rollercoaster, but some things about it are off, which sometimes makes it feel like a chore to sit down and watch. I still think it's great though. (8/10)