Yes, they can all get worms, and they won't all necessarily show symptoms at the same time, either. The worm eggs are spread in the droppings and then the birds ingest them, eventually. Capillary worms kill. They are very hard to get rid of. They are also very common; I believe more common in birds than tapeworm. Safeguard does work, but I like to alternate dewormers since we had the Capillary worm problem in our flock, so that the worms do not have a chance to develop as much resistance. . Valbazen is good too...so is Levamisole. Strike 3 is a good alternate. You should be able to get Safeguard suspension from your vet, as well as in the feed stores. If we use Safeguard, I like the suspension, rather than the paste, but it's a matter of preference. Any broad spectrum dewormer..just make sure it will address capillary worms, especially. Many dewormers will not kill tapeworm, though.
The worms wouldn't cause the air saculitis directly, but they can lower immune resistance which opens up the door to all kinds of things. And they will kill the birds on their own, if you don't deworm. Guaranteed.
good that she is eating and drinking, that is always a very good sign. You can offer her a little special food mixed with her feed to tempt her further; our sick birds like applesauce, and small soft breadcrumbs made very damp with buttermilk. Yogurt and scrambled or chopped hard boiled egg can tempt sick birds with poor appetite, too. Or chopped tomato or grapes. There's a whole list of stuff you can try.