Not illness but injury and sort of happy ending. Gen
Master of None http://www.fanfiction.net/s/4930428/1/

Amnesiac-Sam, Gen
Ghosts in the Fog http://www.fanfiction.net/s/3035552/1/

Fusion ''verse. Post-Hell Sam with some mental issues. Bittersweet.
http://ratherastory.livejournal.com/102813.html

This one I actually haven't finished reading yet so I don't know if it ends happy or not but I think it does.
Home Is Where the Heart Is http://www.fanfiction.net/s/5937062/1/

Sam is lost in the dessert and gets injured/ill. Loved this one.
The Flight Of the Impala http://www.fanfiction.net/s/5712897/1/

I'm not certain if this one quite fits the category or not, but it's one of my faves so far (it's unfinished but is still active). Sam's cursed to change back and forth from a wolf (does that count as an illness? lol!) and parts of it really wring my heart.
If You Could Only See
http://mikiya2200.livejournal.com/tag/au%3A%20unleashed

Sort of? It deals with Sam's blood addiction withdrawals and is both heart breaking and sweet.
A Little Too Much Nothing
http://mikiya2200.livejournal.com/5295.html