"Only I eat fish when I'm sick" means "Only I, and nobody else, eat fish when I'm sick". In this case, only is an adjective, qualifying the pronoun which directly follows it, I.

"I only eat fish when I'm sick" means "I only eat fish when I'm sick. That is the only thing I do with fish when I'm sick". In this case, only is an adverb, qualifying the verb which directly follows it, eat.

"I eat only fish when I'm sick" means "I eat only fish when I'm sick; I eat nothing else beside fish". In this case, only is an adjective, qualifying the noun which directly follows it, fish.

"I eat fish only when I'm sick" means "I eat fish only when I'm sick; when I am not sick, I do not eat fish." In this case, only is an adverb, qualifying the adverbial clause which directly follows it.

"Only I eat fish when I'm sick" means "Only I, and nobody else, eat fish when I'm sick". In this case, only is an adjective, qualifying the pronoun which directly follows it, I.

"I only eat fish when I'm sick" means "I only eat fish when I'm sick. That is the only thing I do with fish when I'm sick". In this case, only is an adverb, qualifying the verb which directly follows it, eat.

"I eat only fish when I'm sick" means "I eat only fish when I'm sick; I eat nothing else beside fish". In this case, only is an adjective, qualifying the noun which directly follows it, fish.

"I eat fish only when I'm sick" means "I eat fish only when I'm sick; when I am not sick, I do not eat fish." In this case, only is an adverb, qualifying the adverbial clause which directly follows it.