Hi Jeffrey and
to the forums!
You've come to a great place to learn even more about budgies!
Is that the breeding cage?? If so, it is much
too small for any
bird, especially a breeding pair with chicks. Additionally, round cages can be traumatic for birds as they have no corners with which to build their spatial orientation of their environment.
It's very likely that the lack of space is contributing to the female's stress, which may in turn cause her to sit more tightly on her chicks.
I'm glad that the third chick is doing well so far. I truly believe that the best way to prevent this chick from also dying is to ensure that they are being fed a good breeding diet, which it sounds like you've offered them, and are in a large enough cage so that they can move around and de-stress. Once the chicks start to get feathers, you should move the little family into a larger cage, if possible. To minimize stress, try to arrange everything in the new cage in the same general orientation as it was in the round cage.
You should not put any cotton nesting material. This actually should never be used for birds as it is extremely harmful if ingested, as it can cause crop impaction, which is often fatal. Additionally, the mother or chicks could easily get tangled in it.
It is normal for the mother to kick the shavings out of the box until the chicks are born. She should have at least a little shavings in there now to absorb the chick's droppings, etc.
Be sure to read through all of the many articles and stickies that we have here! If you have any questions after reading through everything, be sure to ask as we'd love to help.
Please keep us posted on how things go!