I love me some wee!chesters, so here are a couple of recs that might suit you! They may not fit all of your requirements, but from what you say I think you'll like them. :-)

You should definitely check out the fics of brightly_lit; she writes GREAT wee!chesters! Here are a couple to start with.

Sam and Dean have to escape from a foster home and end up trapped in a blizzard, and Dean has to face how Sam really feels about their life and his father. It’s called Stuck: http://archiveofourown.org/works/680803

John is not absent in this one, but it fits the themes you like, and it has a very little Sam as its focus. He’s four. The First Time Sam Ran Away: http://brightly-lit.livejournal.com/25755.html

And one where John never comes back, and the boys have to learn how to live on their own, What Was and What Could Have Been: http://archiveofourown.org/works/1237045

Brightly_lit has lots of wonderful, angsty wee!chesters, so I might be missing some that fit these themes. Here is her master fic list:
http://brightly-lit.livejournal.com/

To wrap things up, here’s a self-rec—Sam is not sick in it, per se, but it’s wee!chesters, Sam-focused and takes place in a long absence of John’s. It’s called Mother’s Here: http://septembers-coda.livejournal.com/1890.html

Enjoy!

What Was and What Could Have Been (http://brightly-lit.livejournal.com/41533.html) by brightly_lit: "When John doesn't return from a hunt, Sam and Dean, ages 10 and 14, have to find a way to live on their own." (gen)

Side English (http://flawsrevenge.livejournal.com/5845.html) by flawedamethyst: John's gone; Dean learns to hustle pool to take care of Sam. (gen)

Crown and Anchor Me (http://archiveofourown.org/works/133815) by sena: "Sam Winchester is fifteen years old, at yet another new high school in yet another state, he doesn't get along with his distant, distracted father, he's figuring out that he likes guys just as much as he likes girls, his clothes never fit and his limbs ache at the joint ever since his growth spurt started, he has to study for the PSAT and, oh yeah, he's a little bit in love with his brother, Dean, who's taken a break from hunting monsters to work at a local garage for minimum wage." (sam/dean)

With sick!sam

Sins of the Father (http://archiveofourown.org/works/2539502/chapters/5645840) by jcrgirl: "Sam get hurt during a hunt and Dean comes to realize some things about his feelings for his brother and some ugly truths about his father." (sam/dean)

Country of the Heart (http://ratherastory.livejournal.com/194986.html) by ratherastory: "After being hospitalized for meningitis, fifteen-year-old Sam finds himself put in a foster home while John is investigated for suspected abuse. Against all of Sam's expectations, his foster family turns out to be a caring couple with the kind of home Sam only ever saw from the outside while he was growing up, making him feel even more conflicted about the kind of life his own family leads. To make matters worse, John leaves town to go on a routine hunt in spite of his upcoming custody hearing, prompting Sam to wonder if his own father and brother might be relieved to have him —and his constant demands for normalcy— out of the picture. When his father goes missing, followed by Dean when he goes after him, Sam quickly discovers that they need him just as much as he needs them, and sets out to bring his family back together again, for good." (gen - this is one of my favorite stories)

Settle Down My Shivered Bones (http://archiveofourown.org/works/3096767) by quickreaver: "Sometimes John hunts without his boys, and sometimes he stays gone too long. His sons come up with a way to cope. It's not necessarily pretty, but it's what they do." (gen with hints of sam/dean, warnings for eating disorders)

I hope self-recs are ok. Here's one I co-wrote with my bestie. It's weechesters.

More Than Meets The Eye (http://archiveofourown.org/works/864925)

John leaves his sons with Bobby for the summer while he continues to search for the thing that took his Mary from them. Adorableness transpires. Dean is awesome…and Bobby’s pretty good at it too. (Need familiarity with 6.17 “My Heart Will Go On” as fic is set in AU world made canon in that episode.)
Warnings: none beyond unadulterated schmoop; also some h/c and angst, sick!Sammy
Characters: 9-yr-old Dean, 5-yr-old Sam, Bobby, John and others
Co-written with platinumrosel