So what I'm hearing is not "a show like Welcome to the NHK" but rather a show with:
1: Trauma in the character's backstory
2: Character development (probably due to having to deal with the traumatic backstory)
3: Romance
4: Comedy would be a nice plus, but not to the point of overwhelming the drama
5: No ecchi
6: Action is okay
Based on that criteria I'd suggest the following anime:
(Spoiled because just recommending them here might spoil a few things.)
Spoiler for Suggestions:
Fits the criteria very well:
Les Miserables (part of the World Masterpiece series)
Cross Game Trauma, Romance, Comedy - it's perfect for what you want.
Clannad especially After Story (the second season)
Crest of the Stars including the sequels up to Banner of the Stars III A good space opera with everything you asked for. Romance is more developed in the later seasons.
Ano Hana Childhood friends drift apart after a traumatic summer. They come back together in their highschool years to reconcile.
Angel Beats Difficult to describe, sounds like you would enjoy it though.
Starblazers aka Space Battleship Yamato Old space opera from the '70s. Has everything you asked for.
Fullmetal Alchemist: Brotherhood Strong action drama about sin and redemption, with traumatic backstory, character growth, a bit of romance, and small comedic moments.
Maison Ikkoku A romantic comedy about a young man making himself worthy of his love interest, while she has to overcome her grief at her husband's death.
Kimagure Orange Road A romantic comedy set in highschool.
Honey and Clover A dramatic romance set in college.
Neon Genesis Evangelion Apocalyptic, the hedgehog dilema.
Penguindrum Symbolism abounds, apocalyptic.
Touch More Adachi (he did Cross Game as well)
Katanagatari Traumatic backstory, rather verbose action show.
Anime that are pretty close:
Full Metal Panic A bit heavy on the comedy and action, but does have a traumatic background, romance, and character development
Hanasaku Iroha Watch the first episode, a bit spotty because it's a slice of life, but seems to be around the theme you want.
Ah! My Goddess: The Movie You might need to watch the series first to understand though. The series is good but a little more light and fluffy then what you seem to want.
The Melancholy of Haruhi Suzumiya Has a lot of comedy and otaku antics, but does also have the elements you are looking for. The movie The Disappearance of Haruhi Suzumiya is even closer to what you want, but it won't make sense without watching the series first.
Chrome Shelled Regios It's all there, but it isn't fully developed in the anime (the light novels handle it better).
Mirai Nikki Certainly has it all, but is more focused on the action and crazy Yandere goodness.
Tenchi Muyo
Martian Successor Nadesico
Seikimatsu Occult Gakuin
Chrono Crusade Traumatic backstory, romantic subtext, lots of action