Have your birds been to the vet lately? It's a good idea to have them checked occasionally for parasites and such.
Being fluffed and dull is not a sign of good health. So Kiwi is fluffed and Luna has a peeling cere. How long have you had them in that cage? Did you clean it well before putting them in there? What did you use? I'm sorry I can't help you more with this. Anytime one of our pets looks "off" it's best to have them checked by a professional. I know there was some concern not long ago about certain bird cages being toxic. Hopefully they've all been pulled off the market by now.
As for training budgies, there must be a book at the library or book store on taming/training them. It's got to be a lot different (and easier ) than training a Great Horned Owl or Vulture (that's the type of birds I work with).
I've had budgies for over 10 years now and have learned a lot about their general heath and habits.
The first thing I would tell you is that a fluffed up bird is not a healthy bird. Birds in general will not show the signs of illness until they are already pretty sick, because as a creature of prey, it is instinctive for them to "put up the pretense' that they are strong and healthy. A bird can fluff up for so many different health issues it would not be in your best interest to try and self diagnose or medicate it. The fact that they have reversed roles, may be a sign that the more healthy bird is taking over.
The post above by Ireneo brings up a good point, if you have just recently noticed this behavior since changing the cage, something the cage is made from could be toxic. Was it a new cage or did you get it somewhere and clean it yourself? If it's a second hand cage a sick bird could have used it and it will still have disease. What do you clean the cage with? A cage should not be cleaned with any kind of common household cleaners as they are toxic. There are non toxic cage cleaners you can buy from a pet store, organic are good.
As for the birds "cere" which is the fleshy part above the beak, it can change color and texture for many reasons. In a female budgie, it will turn a brownish color when the bird is getting hormonal and is ready to start breeding. It depends on the color of your bird what color it will change to but as a rule, it will get darker and brownish colored. The female cere is usually a fleshtone to a beige color, and the male's are blue or almost purple blue depending on the birds plumage. If you have a very light colored male, the cere can also be flesh toned, but will have a blueish cast to it. I've had some recessives that had pinkish blue ceres that even the vet thought were female but turned out to be male.
In any case I would bring the birds to a vet for a through exam, and save yourself a lot of heartache as a sick bird will go down hill pretty fast. You can find an avian vet in your area by using a google search or asking local vet to recommend one. Don't make the mistake of taking it to a vet that isn't trained in avian medicine. I learned this the hard way and lost a beautiful boy to a vet not giving him the correct medicine for a respiratory illness years ago.
Training a budgie is a challenge at times, but they are very smart little birds and they will get used to a schedule if you follow a few steps. If you let them out of the cage for some exercise, you need to have them in a very safe environment.
This means no mirrors or windows they can crash into, as they cannot tell the difference between glass and air. Obviously no open window, but also no areas where they can fly into a bathroom or kitchen and fall into any water, as in the toilet or into a pot on the stove. They should also be kept away from any houseplants that may be toxic. Budgies enjoy nibbling on greens, and should be given a variety of dark green leafy lettuces and the like in addition to their regular diet. That is another thing to consider, a budgie should not be give a seed only diet, it's too high in fat and they can develop tumors.
A very important thing also is never use a teflon or non-stick pan in the area your bird is as they are highly toxic at high temps and can kill the bird.
There are many good books that are out their for budgie care and training, you may even be able to find one at your local library.
The first thing I would advise you to do is to get them to an avian vet and tell them about the cage. Until you get the bird there, I would put it back in the other cage and keep it warm and quiet. If you have a hot water bottle you can wrap it in a towel and put in in the bottom of the cage. Birds that are sick have trouble maintaining their body temperature and need to be kept warm.
Make sure the bird doesn't get overheated, just keep an eye on it and if it's raising it's wings out from it's side it's too hot. Never let the birds get a draft from an air conditioner or a fan either, where is your cage placed?
I hope this helps you and please let me know how the budgies are doing.