What to do about my finches losing feathers?

I have two finches: a young Cordon Bleu and a Society. I've been noticing feathers from the Cordon on the ground, but as I was clipping the Society's nails today, I noticed that it was losing feathers too. I've sprayed them with mite spray, and they have a birdbath, seed food, fresh water, and a clean cage. What do I do? Should I separate them? And if you can tell me how to clip their beaks too, that'd be great. Thank you!