If you don’t like your cousin, you should write:
I don’t like my cousin.
Seriously. You don’t need to replace ‘like’ with fancy words, especially when you’re not sure of the meaning. Just because a sentence is simple, it doesn’t make it bad.
Today I came across this sentence: ‘I didn’t favour my cousin.’ From context, that didn’t really make sense. Someone was asking the narrator to share a room with said cousin, whom the narrator obviously dislikes.
You can check the definitions of ‘favour‘ and none of them actually mean ‘like’. The closest meaning that sentence can take for it to still be somewhat grammatically correct is if it is saying, ‘I didn’t treat my cousin any better [than anyone else]’, or ‘I didn’t prefer my cousin [over someone else]’, or even ‘I didn’t resemble my cousin.’ None of those is what the writer meant.
There’s a lot of writing advice floating out there about using this word instead of that. Sure, some of it is useful and adds more variety to your writing. Often, though, it’s best to be simple and direct.