Hi Snigdha,
I am so sorry this is happening to your birds. I do understand your limited resources. Unfortunately we cannot even begin to guess what the problem is. What you describe are symptoms that any bird will exhibit when they are gravely ill and dying. There can be many causes. When birds in the same flock or location keep dying, some causes can be a virus, bacteria, fungus, other diseases possibly transported by insects, contaminated water or food, some type of toxin in the air – as you can see it isn’t possible to know the cause without testing by a Veterinarian.
Air sac mites are very common with Gouldian finches. Air sac mites are believed to be found in most if not all captive and wild Gouldian finches and a symbiotic relationship is suspected. They mostly succumb to these mites during molt, stress, extreme temperatures or if they have poor nutrition. Preventative treatment needs to be administered at the right time, and not year round. The treatment itself can be toxic. If there is a Vet in your area who treats birds, they would know what medication to give. We cannot legally try to diagnose your birds or recommend any medications, unfortunately.
I hope you can find someone in your area who can help. If there is a breeder of Gouldian finches in your area, maybe they can help you. I would suggest contacting who you purchased these birds from. Again, I wish I could be of more help but we can only recommend health issues to a Veterinarian.
Take care,
Brenda