Gosh that's tiny.
Okay...well, to start with, although 10 gals is certainly adequate space size-wise, it is very hard to adequately heat-regulate a tank that size. Baby beardies need somewhat higher temps than adults can be happy with: 105-115 in the hottest basking spot right under the light and 80-85 ambient temps throughout the tank. Temps can go lower at night, but I do like to use a low-output undertank heater for night heat for fragile animals. Getting a basking spot up that hot makes the rest of a small tank too hot, too hot to be able to cool down when he wants. 90 degrees is not quite warm enough for a baby.
UV lighting is very important. Not only is light brightness important to appetite and feeding response, but they need the UV lighting to use calcium. Without it they are at risk for metabolic and deficiency disorders.
Soaking is good, like you are doing, to keep him hydrated. The lack of eating can certainly be due to the stress of moving. A baby so young (doesn't make me think much of the breeder for selling them so tiny) has no reserves to tide him over until acclimated, so he has emaciated pretty quickly.
Also, you did not mention any live foods (maybe it was an oversight?). Babies and juveniles must eat far more protein foods than they do veggies. What was the breeder feeding them? You need to get some tiny crickets (and I mean tiny, no bigger than the space between the dragons' eyes), feed them (gutload mix and the same veggies as your dragons) for 24 hours, dust them with some good quality reptile calcium and vitamins (Look for no-phosphorous brands when using crickets)before offering them, and offer them 3-5 times daily. Babies will eat their fill in about 10 minutes and as they grow, will eat more than you ever imagined possible. They won't start adding vegetable foods in any significant amount for another month or so, though you can certainly keep them offered, along with juvenile beardie pellets, daily. They should eat mostly protein foods for the first six months and then still daily for about the first year.
One risk in a baby this young is an anorexic cycle. Loss of appetite leads to such a decrease in metabolism that appetite and feeding response cannot recover and they can starve to death in front of their food.
Here is what I would attempt: first, increase cage size to about 20 gals for now. Make sure the temps right under the basking light are at the upper range and that he also has several hiding spaces throughout both the warm and cooler areas. Get good UV lighting on top. Cover the tank on at least three sides so that he feels more secure. Handle and look at him only when tending to him and try to do everything at once, several times a day. Soak him in warmed Pedialyte. Drop some on his nose. Will he drink? You can mash up moistened baby pellets, thin them with water and drop them also, or you can try chicken baby food thinned the same way. There are also food-replacement solutions available from your vet. Then, try offering a few of the tiny crickets. (Make sure anything he doesn't eat comes out right away. It is usually a good idea to remove cage furnishings for baby feedings, or use a separate container.) If he will eat, don't overload him right away. Let his digestive system build itself up again over several days before you offer a full meal. After that, let them eat all they want in about 10 minutes, again at least 3 times per day.
I would probably also get some probiotics for support of his digestive system. Acidophiliz and Bene-Bac are two brands. A few drops daily (The Acidophiliz is really good) will help.
Be patient, and be prepared. Hopefully he will turn around and regain his appetite and his health. However, keep in mind that some babies are too fragile, or may even suffer from some problem you cannot see, to thrive and live. Being sold so young was not your fault, but the stress of such is too hard on such a young animal.
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4.3 Bearded Dragons (Rescues)
1.0 Corn snake (Rescue)
0.1 California King
4.5 Royal (Ball) Pythons (3 rescues)
1.1 Dogs (Rescues)
2.0 Cats (Rescues)
2.2 Humans
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