You can think of 飲物 as “beverage,” and 欲しい is used to say that you want something (usually that you want a particular thing, and not that you want to do something). So the second sentence literally breaks down as “some kind of beverage is wanted (by me).”
The first sentence emphasizes the verb 飲む (to drink), specifically wanting to drink.
The second sentence emphasizes 飲み物 (a noun meaning drinkable liquids), specifically desiring a drinkable liquid.
It’s the difference between “I kinda want to drink” and “I kinda want a drink”.
Japanese natives tend to be pretty indirect, though, especially when they want to be polite. The more common colloquial expression would probably be something like, 「なんか喉が乾いてますが」(“I’m kinda parched” or, more directly translated, “Say, my throat’s a little dry”).
Note that the なんか at the beginning softens all three sentences somewhat, but it’s a non-sequitur in the last two. In the first sentence it’s the literal object of the sentence: you’re saying you want to drink something (but subjects are almost always optional in Japanese, “something” would still be implied without it).
なんか doesn’t really mean “something” in the last two sentences though. It’s more similar to an American high schooler (“valley girl” if you’re old enough) beginning a sentence with “Like, I wanna drink”, or an old forties-era Hollywood actor uttering, “Say, I’d like a drink”. It’s sort of a conversational opener implying: “I’m about to say something, but it’s not terribly important.”
It sounds somewhat harsh to be too direct in conversational Japanese. None of the sentences change in meaning without なんか, but they would sound somewhat more demanding/jarring/commanding/direct if you didn’t use it.