You could also ask in 'Adventure Time' - related Facebook groups. There are often fans in those groups who are into fanfiction and might be able to point you in the right direction. Additionally, following 'Adventure Time' fanfic writers on Twitter might lead you to their works, especially if they write about sick scenarios in the show.
Try searching on Tumblr. There are a lot of 'Adventure Time' fans on there who might post or recommend sick fanfics. You can use relevant hashtags like #AdventureTimeSickFanfic. Also, some Reddit communities dedicated to 'Adventure Time' might have discussions or links to such fanfics. For example, the r/adventuretime subreddit could be a good place to start looking.