Do you say “I want to drink” or “I want to drink something” when you’re thirsty in everyday’s English? In my native language we don’t say I’m thirsty but just “I want to drink”. But I think in English it may mean that persons wants to drink alcohol or it may not?

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This is a good question.

On the one hand any sentence that includes the word 'drink' can simply refer to consuming a liquid, but on the other hand we often use use 'drink' to refer to alcohol.

We probably wouldn't say "I want to drink" . It's more likely that we say "I want/need a drink" , and this can simply mean "I am thirsty" , but,  as you said, it could mean "I want/need alcohol".

In certain phrases it's clearer. "You drink too much" probably means "You consume too much alcohol", but technically that's not 100% certain.

Therefore it really depends on the context

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"I want a drink" "I want something to drink". "I need to drink something" --> is probably a little better when you have to emphasis the seriousness of your thirst.

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First choice: "I'm thirsty".
However, "I want something thing to drink" is fine and does not give the impression you are looking for alcohol, unless you are in a bar. (Context matters.)

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*person