What Remains of Edith Finch was pretty good!

Thanks for sharing the game!

the sequel to to the moon was in the last monthly, if you haven't played it.

I'd add Her Story, Valiant Hearts (Oh, my feels, you'd love this one if you are into moving stories) and The Lion's Song.
Also, Hellblade is a pretty good story driven game, but not so indie, I guess.

Valley is really nice 'running and jumping simulator' with beautiful landscape and exploration. Word of warning though: game is quite short :/

adding to what other said, if you haven't played them:
Tacoma (same guys that made Gone Home), The Vanishing of Ethan Carter, and pretty much all of Supergiant Games titles (I don't really know about Pyre, but Bastion and Transistor were awesome)

A few point and clicks, or various adventure-type games ( based your examples, anything with story and without combat? goes I guess)
J.U.L.I.A.: Among the Stars
Tavernier
Unavowed
Yesterday (and Yesterday Origins)
A Normal Lost Phone
Alicemare
LiEat
Another Lost Phone: Laura's Story
(While it's a metroidvania, I do recommend Aquaria. Combat is simple and sadly mandatory, but it has awesome lore to it, and it's both beautiful and haunting, exploring the world alone, what happened before)
Bad Dream: Coma
All of the Blackwell games
Brink of Consciousness: Dorian Gray Syndrome and Brink of Consciousness: The Lonely Hearts Murders are awesome crime/thriller HOGs. Great story, suspense, good production values. Best hidden object games IMO, without the sugarcoaty-cliché Mundi usually uses.
Brothers - A Tale of Two Sons
Contradiction - the all-video murder mystery adventure (Highly recommended if you love FMVs. Game is mostly watching videos and thinking though)
Dead Synchronicity: Tomorrow Comes Today
Disturbed (free)
Evening Surprise (free)
Forgotten, Not Lost - A Kinetic Novel
Hector (the telltale game)
Her Story
Highway Blossoms
How To Cope With Boredom and Loneliness (Free, highly recommended)
Hustle Cat
Jazzpunk: Director's Cut
Just, Bearly (free)
Marie's Room (free)
MechaNika (YES YES YES)
Memoranda (ALSO YES YES YES. Consider using guides if it gets too hard. It's too good all-around to spoil it with struggling with the gameplay)
Message Quest
Old Man's Journey ( Very much recommended)
Orwell
Perceptions of the Dead (free)
Princess Remedy in a World of Hurt (free, shmup-like fights otherwise it has a great story/setting/tone)
Residue: Final Cut ( Decide it for yourself. A weird game and not super polished, but has such a unique topic to it, it's one of the most memorables for me)
Any Rusty Lake game
Serena (free)
Shady Brook - A Dark Mystery Text Adventure ( Looks a bit like ass, but if you like Twin Peaks-y "something is shady here" mystery story, this is fucking amazing)
Space Pilgrim 1-4. Surprisingly competent story, the games play like point and clicks. No fighting.
Stick It To The Man! (very much recommended)
Stray Cat Crossing (very-very much recommended. If you liked The Cat Lady then this should be up your alley)
The Bottom of the Well (free)
The Darkside Detective
The Dig
The Little Acre
The Monster Inside (free)
Why Am I Dead At Sea (very-very much recommended)

I think that's all from my completed game list ^^ I hope you can find something interesting

Beyond Eyes is probably the most suitable one I can think of. I also enjoyed The Charnel House Trilogy, Oxenfree and Hacknet.
The Grandfather is definitely a small oddity but not at all endearing.

Oneshot
Machinarium
The Silent Age
Tsioque
...

Check out:
Brothers - A Tale of Two Sons
Blackwell series
Dreamfall Chapters
Jazzpunk
Night in the Woods
Primordia

The Lion's Song is a nice experience, or at least it was for me. Also, Last Day of June (next Epic free game) and Stories: The Path of Destinies I would suggest. Old Man's Journey and Burly Men at Sea are also stories disguised as games. I don't know if they are defined as story driven - but I liked them, and their stories.

Some people already mentioned it but Brothers - A Tale of Two Sons should be right up your alley.

Rakuen is wonderful, and reminded me a lot of To the Moon.

Also:

A Bird Story
Last Day of June
Old Man’s Journey

Cheers for the wishlisted giveaway, here are some games I've really enjoyed:

story-driven indie games

• Marie's Room – An excellent and very short free game, I've not not played more than a few minutes of Gone Home but it seems similar in style and tone
• RiME – Beautiful 3D exploration puzzle game, with a story that's deeper than it first appears

interesting and endearing oddities

• Refunct – very short first person puzzle game
• Off Peak (free) and The Norwood Suite – Two completely bizarre yet brilliant "walking simulators" from a guy called Cosmo D
• The Haunted Island, a Frog Detective Game – Another short game (I won it here), it's incredibly simple both in visual style and gameplay, but quite charming ;)
• The Stanley Parable – A brilliantly funny game about games
• Dr. Langeskov, The Tiger, and The Terribly Cursed Emerald: A Whirlwind Heist (free) – Another excellent commentary on gaming from the guy behind The Stanley Parable

Gone Fireflies is a very small and short, but really good game. I'm pretty sure you'd like it.

Draugen is a short, yet beautiful game with a great story!

Night in the Woods is one of my most favourite narrative-driven games in recent years, heavily recommend it!

Tacoma - The next game from the Gone Home devs, if you haven't played it yet
The Red Strings Club - My favourite game from last year. Really engaging cyberpunk narrative that's not afraid to ask some difficult questions
ICEY - A hack and slash which takes a Stanley Parable kind of approach to its narrative, leading to some very interesting meta stuff
The Novelist - A game where you're a ghost watching over a family and trying to influence their decisions so that they don't fall apart

Nice Sunn O))) avatar btw ^_^

ty

Anything by Amanita has been great so far! Samorost, Machinarium, Botanicula, etc

Thanks~

You should play The Wolf Among Us.

Still Life 1 and 2

I hope you don't mind classics as well:
Almost any Wadjet Eye game (especially Unavowed)
Downfall (I haven't played Lorelai yet, still waiting for it to be free of bugs)
Gabriel Knight 1-3
Dreamfall saga (especially The Longest Journey)
Fran Bow
Shannara, Death Gate, John Saul's Blackstone Chronicles (maybe also Companions of Xanth)
Night in the Woods
Firewatch
Grim Fandango
Oxenfree
The Wolf Among Us, Tales from the Borderlands, The Walking Dead (the first one, of course)
The X-Files
Still Life (not the sequel!)
Culpa Innata
Blade Runner
The Last Door (both of them)
The Vanishing of Ethan Carter
Black Dahlia
Tex Murphy series
Memoria
Draugen
Observer
Contradiction - Spot The Liar!, The Shapeshifting Detective, The Infectious Madness of Doctor Dekker

Recent and pretty awesome indie adventures almost nobody have played yet: Detective Di, Rainswept, Whispers of a Machine.
You might also enjoy Tale of Tales art (only the last thing they've released, Sunset, can be actually called a game, perhaps); at least give The Path a try, if you haven't yet.

Dreaming Sarah is a very short game that's in bundles all the time, but it's exactly the kind of thing you're asking for and it looks like you don't already own it.