I think any fish will do that. It indicates that things are in a very bad way.
I’m not aware of any structure that goldfish have that would make them better at breathing air. The Anabantidae have an organ that lets them breathe, corydoras take a gulp of air once and a while although they fart it out quickly. 
Sorry, I’ve gone into Aquarist Geek mode.
So OP, you’re filtering the water like hell and have many plants in the tank, right? :dubious:
Carp are supposed to be better at it. They’re generally hardier than tropical fish, and can survive worse conditions, but it’s a mistake to think they can survive or thrive in low quality water. They’ll just stay alive a little longer than other fish. And they do make a mess. When changing water, a gravel vacuum cleaner should be used to siphon out water and clean the bottom of the tank. I had to maintain a tank full of feeder goldfish to feed my grouper and triggerfish. Filthiest fish I’ve ever seen kept in a tank. But from the reactions I saw, goldfish must taste much better than guppies.
I stand corrected.
The major cause of death of large cichlids is supposedly “goldfish ingestion”; the large heads stick in their gut.
Hadn’t heard of that, but I didn’t keep cichlids (good lord it’s been 20 years since I kept any fish!). I wouldn’t be surprised if a variety of fish would have a problem eating goldfish. As they get bigger they get more solid. Picasso (the triggerfish) would disable them with a gut shot, then pick them apart. Grouper (the grouper) would inhale them whole. He certainly was built to do that. And moments later he would spit out a cloud of goldfish scales, so there was another mess to clean up.
BTW: I’d recommend reverse flow undergravel filters for any tank with a gravel bottom. The worst messes occur in the poorly oxygenated water in the gravel bed, and the usual undergravel filters that draw water down through the gravel just pull all the muck into the gravel. For saltwater a protein trap is also excellent at maintaining water condition (never tried it with fresh). And although it was anathema outside of invert tanks, algae growth was very good also.
Sounds fishy to me.
It has been determined that ammonia is in the ‘‘stress’’ level which brings up several other questions. Why is the ammonia in similar aquariums not elevated? Why does only the largest fish show lethargy?
Goldfish poop more and produce more nitrates. I am not an ichythologist, but presumably larger fish take in more nitrates/ammonia and are affected first.
I was deleting my deleted email…How did things turn out, Janesemail?