Am I correct is assuming that the two birds you have now are cockatiels? Were they hand reared or raised by their parents?
As suggested, a local vet that is an avian vet may know of a way of placing one of your birds with a good family. It sounds like the two of them may have bonded with each other so they are not receptive to you. I'm not sure what separating them will do. I would hope that in time the remaining bird would bond with you because they are very social and need attention. You might ask the vet about what to expect if you separate them.
I've always had only one bird at a time. I would hope that with time you can earn the bird's trust, and the biting behavior will cease. I wish I could be of more help.
p.s. Our cockatiel really feared some ringtones. When the phone rang he would fly away in a panic, and he actually hurt himself quite seriously one time. We finally found a phone that had a ringer he would tolerate. I don't know if it was the pitch that bothered him. He also hated the warning sound for the emergency broadcast system that would come up on the TV .
p.s.s. Try offering your birds cooked rice, scrambled eggs, cooked pasta, cheese, cereal like cheerios, any type of cracker (but watch the salt), bread, cooked cereal like oatmeal, fruits, and nuts. Avoid avocado and chocolate.