I have found in medical settings it’s best to be blunt. If I gotta vomit I say so, loud and proud.
Usually it’s “look out, I’m gonna puke!”
Yes, there seem to be recurrent cases where slang uses reversal. Like, of course, ‘bad’.
Oh well, usage eventually determines validity in English. We don’t have an Academie Francaise trying to standardise the language… a hopeless rearguard action anyway.
To add - I don’t necessarily associate the phrase “feeling sick” with nausea, unless they say “feeling sick to my stomach.”
When I’m minorly sick, I usually say “I’m not feeling well”, “I’m feeling unwell” or “I’m feeling under the weather”, the last one might be an old people expression. My kid used to call it “the itis”, a reference to the suffix, I presume.
If the illness is not minor, I’m usually like “I just got slammed with the fucking flu” or “stay away from me I’m dying” or “I’m on my deathbed”. Like many people that are rarely seriously I’ll, I can be dramatic when it occurs.
My mother would sometimes use the phrase “feeling punk” for being somewhat ill.
I use “I don’t feel so good” a lot for minor illness.
Feeling puny, is one I’ve heard.
To me, ‘congested and feverish’ implies coughing, fatigue, and body aches.
So I want a word that describes the feeling that goes along with those, but is not actually those symptoms. Like right now I have all those symptoms but while they suck they aren’t the worst thing about whatever bug I have, it’s the how crappy I feel. Like I’d be ok working with just those symptoms (i work remotely so I am not infecting anyone, whether I stay in bed or get up and go to my computer) but the reason I am still in bed is I feel crappy/ill/sick.
I think this probably technically correct, and describes what I’m after but I don’t want to sound like a consumptive 18th century poet 
oh, I was going to suggest malaise. How about “under the weather”?
I like the British “poorly.” But since I’m not British, I can’t use it without sounding pretentious.
Like some others, to me “sick” does not necessarily suggest nausea/vomiting. So I will usually just say “I’m sick” or “I feel sick.”
My mother used to say “Got the epizootic?” when I was feeling ill.
Is this still a thing? I ask because I’ve been watching some recent French police procedurals, and I’ve been amazed at how many English words are used routinely. In one, the title character uses “stop” more often than “arrete” (sp? I think it has that teeppee shaped accent over the first ‘e,’ too).
I’m from Britain (the South/Midlands) but I’ve been in the US for twenty years (mainly on the West Coast)
I do use this term. But I wouldn’t use it unironically about an adult, like I’ll use it to my five year old whenever she gets sick (she is the primary plague vector in our household as she just started kindergarten)
I mean it’s always been the case in my experience, that the one thing the French like more than coming up with arcane bureaucratic regulations is, ignoring their arcane bureaucratic regulations;)
“Feel like I been shot at and missed, shit at and hit.” 
A purist would say that you can only be nauseated. If you are nauseous, you’re actually causing nausea in someone.
That’s from a movie with Robert Duvall, actually.
My father, a gastroenterologist, was a stickler on those two words.
It seems to still exist, according to Wikipedia at least. How much authority it has, or how much attention is paid to it in practise, I can’t say.
Watching these shows, not much.
Or a Pink Floyd lyricist. 