Goldfish can get some oxygen from air, but the gasping is mainly because gas exchange in water takes place at the surface, so that's the best level of oxygen if levels are low.
Changing all the water daily is not necessarily bad. So long as the new water is dechlorinated and the same temperature as what they had before, new water is better than ammonia build up, which damages gills and shortens lives, or worse.
If you know anyone with a working aquarium, you might be able to get some used media from their filter, and use that to cycle a new tank very quickly. Bypass the time needed for the more usual cycling methods. You can also use a lot of plants to help cycling, but they will eat many of them, and that's messy.
Best thing is the new tank, but goldfish do grow, and live a long, long time. They'll outgrow even a twenty gallon fairly soon. Rehoming them may be a better idea in the long run, if you can find someone with a pond that will take them.
The bubbler may be creating a bit too much current, but they are hardy fish. If you can, lengthen the air hose, the longer it is, the less air reaches the stone, and slows it down. Might have to add ten feet to the hose. You could also pinch the hose a bit, say, a clothespin, to reduce the air flow.
Edit. I like that you trying do the right thing here.. but goldfish, despite the many myths and misconceptions about them, remain popular, even though they are not good tank pets. They are better off in ponds, IMO
, because they do get too big for most tanks within a short time.
After 3 years in a bowl, yours may in fact have been stunted a bit. This can happen if they are kept too long in too small a container. If so, it's permanent, but I doubt they are in pain from it, if that helps. But they would still do better in a pond for the long term. It is just a sad fact of life, and pet stores don't tell you that bowls are a bad idea or that the fish need as much space as they really do need. Not your fault, so try not to beat yourself up over it. You are trying now, that's what counts. I'm a life long animal crazy lover too, I hear where you are coming from.