Your translation is close, but even though it is in the same sentence as
お酒直飲みしたくて, it is actually an idea that might be best translated into its own thought (though still part of the same sentence using a conjunction). Let's start with how these are connected:
S/he only wants to drink alcohol directly from the bottle, and...
(The subject of the sentence could be anything, and 'you' could work just as well. Without context I'm shooting in the dark, so I'm choosing the shortest option for translation.)
The part I have in bold is the key to this conjunction. I don't know if in your studies you have learned much about translating English conjunctions, but this is one of the Japanese equivalents. Simply turn the last word in the sentence to the base-て form. For the most part, I would treat it as an 'and' in translation, but there are exceptions.
So now that we've gone through the first phrase of this conjoined sentence, let's look at the second phrase:
... they always gulp it down like that.
いつも is translated to 'always' in this sentence, which is pretty much the main translation. You could also say 'every time', but I think it's a matter of personal preference in translation. I don't think it changes much.
そんなに in this case translates to 'like that'.
呷って as you have aptly translated means 'gulp down'.
さ is a sentence ending particle that is similar to よ in meaning, but it tends to be overused like the word 'like' in English. As such, it really doesn't carry much meaning in the sentence except to add some emphasis to the phrase. Here's a good source about さ and other sentence ending particles.
Notice that the verb in the second phrase also ends in the base-て form: 呷って. This is, once again, a transition into one more idea. In translation, because of the ellipsis (...) at the end, I would translate it as a long pause without completing the sentence. In all, the first line as the following:
S/he only wants to drink alcohol directly from the bottle, and they always gulp it down like that...
This last phrase is where @chocolate is right to question whether the speaker is native, as the causative form of 怒る is used incorrectly. The speaker should have used を instead of に. According to this source, using に with an intransitive verb (to get angry) is more like saying 'let __ get angry.' That doesn't fit quite right. Using を in this case is like saying 'make __ get angry,' which is more along the lines of what we are looking for.
So finally, the corrected phrase and its translation:
it makes me angry...
Once again the て form... The speaker is leaving the sentence open as if they want to say more, but I doubt that they actually will. In Japanese, a lot of the time, if you leave sentences open, what is left unsaid can carry just as much (if not more) meaning than what is said.