You're speaking with Dr. Michael Salkin. Welcome to JustAnswer. I'm currently typing up my reply.
I'm sorry to hear how uncomfortable your budgie is. You've described a difficult molt and the bits of the keratin (a structural protein) sheath falling away from new (pin) feathers. That sheath needs to be removed for new feathers to fully emerge. In most cases this type of molt indicates a nutrional deficiency - most often in protein and calcium. Nutritional imbalances are a common cause of illness in our pet birds. What has her diet consisted of, please? Seeds should compose less than 20% of her diet. A diet of mainly seed and nuts has excessive fat, carbohydrates, and phosphorus; marginal protein; adequate vitamin E, and are deficient in amino acids, calcium, available phosphorus, sodium, manganese, zinc, iron, vitamins A, D3 (necessary for efficient absorption of calcium), K, and B12, riboflavin, pantothenic acid, choline, and available niacin. Ideally, a balanced pelleted diet such as can be found here: www.harrisonsbirdfoods.com or here: www.lafeber.com/pet-birds should be fed as well as hard boiled egg yolk, pancakes and cornbread, the tops of fresh greens, dairy products such as yogurt and cheese, fresh fruits such as apples, pears, melon, kiwi, and berries, vegetables such as broccoli, carrots, beets, asparagus, cabbage, sweet potato, and squash, and even tiny pieces of meat.
I understand that she's being stubborn about eating more than seeds. Her life depends upon doing so, however. Here's a one manner in which transitioning her to pellets might be accomplished: http://www.harrisonsbirdfoods.com/using-our-foods/small-bird-conversion/
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