So sorry @vowled for getting to this so late, but here are the Severitus or Severitus-ish fics that I saved on my AO3!
"Sorry, Aunt Petunia." by Emily_Elizabeth_Fowl, sparklygems
A single change can make a world of difference.
Or, in other words, Harry accidentally says "Sorry, Aunt Petunia," to Snape of all people.
Attached by randomausfulloffluff
Soon before Voldemort kills the Potters, Lily comes to Severus and makes him promise to protect Harry. This vow alerts him when Harry is in danger a week after his parents' deaths. After retrieving Harry to Hogwarts, Severus is determined to keep his distance as much as possible — but when does Severus ever get what he wants?
In which Severus is a Glorified Potter PillowTM and the Hogwarts staff is the peanut gallery.
Live and learn by Hejsan_Svejsan_Booo
Severus never imagined his life would be like this, everything changed when he found a young hurt girl in the woods who he decided to take in and call his daughter. Thanks to her he changed his whole trajectory of life, and now here he was, trying to save the Longbottom’s. His daughter befriending Percy Weasley, and leading Severus straight to a young and hurt Harry Potter. But even though his life was chaotic he would not want to change it.
AU. Take one Severus Snape, one baby Potter, and one insane Headmaster; simmer gently and stir well. Don't forget to duck!
More Than One Way to Skin a Cat by AverageFish
Bleeding out on the floorboards, Severus greeted death as a friend.
He wasn't expecting to wake up.
Stuck in the infant body of Harry Potter, with his addled cousin by his side, Severus is walking a mile in a very different pair of shoes.
A time travel fix-it. The oddest Severitus (Sevitus) you'll ever read. Betad by Eider Down.
Stripped Down and Rebuilt by SofiaDragon
Severus Snape doesn't think Albus' plan to martyr Potter will work out and knows nothing of the secret setup to give Harry a chance to live through the killing curse a second time. He decides that what they really needed was to start hunting the Horcruxes years ago instead of waiting for children to do it for them. He sends back his private notes and everything needed to brew a potion to strip off the spells that are restricting his behavior. He expects that his younger self will receive the information in 1992 at best.
The adventures of Snape slowly adopting Harry Potter by Flumet
A series of one-shots that follow Severus as he first takes care of Harry "temporarily" and then slowly becomes his adoptive father.
After the death of the Dursleys before Harry’s eighth birthday, it’s suddenly up to Severus Snape to take care of him. Temporarily, of course. Until a better guardian can be found. Severitis (non-biological). NO corporal punishment. AU, but follows canon knowledge of the Dursleys.
The Tragedy of Severus Snape (as Directed by Teenagers) by destieldumbbitch
All Harry wanted in his third year was a bit of peace, maybe some decent marks in Potions, and definitely fewer life-threatening disasters. Instead, he somehow becomes Hogwarts’ unwilling therapist, accidental matchmaker, and (apparently) Snape’s semi-adopted son. Featuring: a gleeful matchmaking conspiracy, Lupin deciding subtlety is optional, Snape suffering in silence (mostly), and far too much student gossip.
There's Something Different About Snape by a_decently_sized_worm @a-decently-sized-worm
The year is 2025. An average guy gets hit by a car on his way home from work and dies.
In another universe, the year is 1992. Severus Snape wakes up. He proceeds to have a panic attack.
What if? by alwayslily22, and Des98
Baby Harry is adopted by McGonagall and Pomfrey and Severus Snape is wrapped around his little fingers.