Children Of An Elder God: Subverted. At the beginning it looks lighter and softer than canon: the characters are a bit happier and more psychologically stable, the battles against the Angels are less traumatic... and then more and more elements of the Cthulhu Mythos are introduced, and the setting becomes very dark.
Last Child of Krypton: In this setting Shinji gets Superman powers and a lesson on power and responsibility. He grows into a more idealistic, more caring and less passive person, and that different mindset and their powers let him alter events positively.
Superwomen of Eva 2: Lone Heir of Krypton: To canon. Asuka gets Kryptonian powers and starts out a super-heroine career. In the process she learns to be a better person, finds a cause to worth to fight for, hooks up with Shinji, and manages to avert some canon tragedies (such like Touji getting maimed or Rei blowing herself up)
The Bridge, a crossover between My Little Pony: Friendship Is Magic and Godzilla has an odd example. While it will deal with plenty of darker moments which are much more grim than the first series usually delves into, the tone and protagonists from the kaiju side are all played much lighter than the norm for Godzilla fanfiction, such as using one of the outright heroic incarnations of Godzilla instead of a True Neutral or Chaotic Neutral version. Compared to other crossovers of the same type such as "Born of Rage" and "My Little Destoroyah", it's much lighter.
The first Ponies The Anthology featured a content warning and even a separate clean version, while the second one was specifically designed to stay within a PG-13 level. Word Of God from one of the Anthology's contributors is that this actually made them think more about what was funny versus Vulgar Humor.
A Crown Of Stars: It is lighter and softer than canon and the preceeding fic. Shinji and Asuka are at last given the help, council and healing time they desperately needed, and a chance to recover their families and make their world right.
Advice And Trust: To the canon history. The story changes a specific event, and as a result Shinji and Asuka lean on each other as they endure trauma after trauma rather insolating from each other, breaking down and imploding.
The Child of Love: Although this story got pretty dark sometimes, it is definitely and unapologetically waffier than the original series. Shinji and Asuka have a daughter, which pushes them to try to figure out their relationship, Gendo gets arrested, and Third Impact never happens.
Evangelion 303: To canon. In this Alternate Universe the main characters are still full of issues, but since they are older, they are somewhat more mature and better at dealing with them.
HERZ: The story can get pretty dark, but it is happier than canon. Third Impact was averted in 2015, and Asuka and Misato survived that battle. Twelve years later Shinji and Asuka are married, have a daughter, and the two of them together with Rei work under Misato's orders to fight SEELE's attempts to cause Instrumentality.
Higher Learning: To canon. A new teacher starts teaching the Children. Their lessons help the three traumatized pilots to consider new possibilities and open up, leading to a better ending.
Once More with Feeling: When he returned to the past, Shinji was determined to set his world right. His -both covert and overt- influence has had good effects so far: Touji's sister did not get crippled when he fought Sachiel, Rei is learning to make friends, Asuka trusts and respects her teammates more than in canon... Nonetheless everything can go to Hell again if he is not careful, and Shinji is well aware of it.
The One I Love Is: A bit lighter than canon thanks to the Love Triangle between the three pilots which led to them finding some happiness and Third Impact being averted.
In the AU Slightly Altered despite having some truly horrific moments, is still a LOT lighter in tone than the original series. As of Chapter 8, nobody apart from Buwaro's biological parents has died, not even Sakido. Rhea is oddly absent, though, so she may be dead.
The rewrite of Sonic X: Dark Chaos is simultaneously both this andDarker and Edgier than the original. While there is more violence and horrific content as a whole, the darkness is balanced with much more likable characters, a stronger message of hope and redemption in the face of terror, and the power of friendship. And Black Comedy.