Even though gapeworms live in the esophagus, some of the eggs still travel the digestive tract and end up in the droppings, so a fecal would likely show them. A vet can also do a swab of the esophagus. You can try a mail in option for a fecal if you don't have a vet that will do one for you...
They have all been exposed. The sick birds likely shed the worm eggs in their droppings before you even knew there was a problem. They can get infected by ingesting those eggs in the droppings (happens while scratching and pecking in the soil), or by eating a secondary host like an earthworm...