For 飲みつつそう言った, I'm confused on how it works because a person can't really say anything while water is in their mouth. However, does that mean the person talked in between sips or does it mean that the person has yet to actually drink anything and simply talked while raising the cup to his lips? I also saw a sentence like 聴衆の前に立ちつつ、新製品の発表を行った, so does it mean that the person already finished drinking and said it since 立ちつつ means the person was already standing, not doing the action while in the process of getting to one's feet?
つつ = ながら
飲みつつそう言った = 飲みながらそう言った
Drink = (to have a glass in hand + carry contents into mouth + put down glass)
Start(have a glass) ----- carry contents into mouth(once or multiple times) ----- End(put down the glass)
Start ---- End
Although you are not drinking when you are talking, what does not stop is not the behavior of drinking. For example, you are drinking as long as the glass is in your hand.
You have to regard the verbs (飲む, 立つ) not as actions, but more like "states of being." 「立ちつつ」 means the person announced the product while in a state of standing. Same with 「飲みつつ」, where "state of drinking" does not begin and end with each sip, but rather continues while the person continues to take sips.
I guess in English, you could similarly say "He/She said so while sipping on their drink," even though, as you point out, it is very hard to say anything with liquid in your mouth!