Oh my, so many problems here, your baby bird is in distress and needs proper help from an avian vet...3 weeks old is way too young for a pet shop to sell you a baby bird!
We need pics desperately because at 3 weeks old your bird won't have a lot of feathers, only pin feathers. So this may be normal, it may not be, I don't know what you mean by "it's feathers on its wing are rotting" but obviously it's not good. But at 3 weeks old I'm not sure what we're looking at.
At 3 weeks old your baby should be feeding regularly and giving you an active feeding response, you shouldn't have to force feed it formula at all. It should regularly take a hand feeding 6+ times a day at this age, and it's crop should completely empty between feedings. So what I'm guessing is that your bird is not in a brooder so it's not nearly warm enough, and it cannot properly digest it's formula because of being too cold. Wrapping the baby in a towel is not nearly enough to keep his body temperature high enough. Without being kept inside a brooder at around 30 degrees Celsius your baby bird will not be warm enough, will not properly digest his formula, will probably not be emptying his crop, and will eventually become very sick. Is his crop empty when you are trying to feed him? As I said your baby bird should be anxiously begging for feedings from you at 3 weeks old, 5-6 times a day or every 3-4 hours, and his crop should be completely empty when it's time for his next feeding. He should have a strong feeding response, bobbing his head and swallowing as you feed him. If you have to force feed a 3 week old baby bird there is something very, very wrong and he needs a certified avian vet immediately. My guess is he's too cold, his crop is not emptying, and you're trying to force feed him while there is still formula in his crop. Without an avian vet he is going to either starve to death or die from a fungal infection in his crop, or from crop stasis, where the formula that is sitting in his crop basically rots and becomes like cement. An avian vet needs to flush his crop out, but to start you need to get him in a brooder immediately!!! He needs to be in one until he is completely covered with at least thick down all over his body. You're saying "his skin is exposed", how much down does he have? This is why we need photos...
Also, what temperature is the baby bird formula when you feed it to him? This is just as important as the brooder temperature, as you need to mix the formula with hot water (NEVER mix the formula with cold water and then microwave the formula, it will have an inconsistent temperature throughout with hot spots that will burn his crop). The temperature of the formula when you feed it to him should be between 105-110 degrees Fahrenheit, no colder and no hotter. If you are even one degree high you will burn his crop and he can die from infection, and if it's any colder it will contribute to the crop stasis he probably already has. Formula that is too cold will not interest a baby, they can reject it simply because it's below 105 degrees. It will also not be digested correctly if it's even one degree colder and will just sit in his crop.
You need to microwave or heat water in a separate container or pan than you're using to hold the formula. Put just as much formula powder as you need for one feeding in a separate bowl (you absolutely CANNOT reheat formula that you made earlier, it is a magnet for bacteria, even if kept in the refrigerator, and you must make fresh each feeding). Mix the hot water into the formula powder, and at 3 weeks you need to make the formula thinner or runnier than for an older baby bird. As he gets older you can make it thicker, but for now make it more on the thin side. You must put a candy or cooking thermometer in the formula after you stir it VERY thoroughly. It will likely be hotter than 110 degrees Fahrenheit at first, so you must let the temperature come down. Since you're only feeding one baby it's much easier to keep the temperature of the formula between 105-110 degrees Fahrenheit, it should stay in that range for the entire time you're feeding him. So be ready with your syringe, towels, etc. and when it gets to 112 degrees get him out of his brooder (that you need to go buy or at the very least build yourself) and when the formula hits 110 fill your oral syringe. I don't know if the pet shop showed you how to hand feed properly, probably not based on their irresponsibility. So I'll go into that next, but pay attention. This is very important and attention to detail is a must. Hand feeding baby birds is not easy at all, it's difficult, a skill you learn over time, and one little mistake will kill your bird. Getting the formula temperature correct is the first very important detail. The second is not feeding him if his crop is still full or has not fully emptied from the last feeding. As I said, at 3 weeks he should be fed every 3-4 hours, so if his crop isn't emptying in 3-4 hours he needs an avian vet immediately because he has a crop infection or is in crop stasis. Also, if you don't have him in a brooder between 86-90 degrees feeding him will be difficult and he won't be digesting the formula. Again, wrapping him in a towel is not good enough, obviously, or you wouldn't be having the problems.
When you go to feed him the thin, properly mixed formula that is between 105-110 degrees Fahrenheit, you must enter his beak from his left side (your right side facing him). You go in the left side of his beak at a diagonal with the end of the syringe pointed at the back of his throat on his right side (your left facing him). Go overtop of his tongue with the syringe. The reason you have to do this is because the opening to his crop is at the back of his throat on the right side (his right). If you don't go in on his left beak, go overtop of his tongue, and aim the syringe at his right side you can very easily aspirate the formula into his lungs, which will kill him instantly. Now a normal, healthy, properly eating 3 week old baby bird will exhibit a very strong feeding response, begging for food, opening his mouth for you voluntarily without you doing a thing, and he will willingly accept the syringe in the left side of his beak and start head bobbing and swallowing as you push the formula into his mouth. Now normally I would tell you to push the formula into his beak very slowly, following the rhythm of his head bobbing. You would understand what I mean if your baby had a feeding response. This is why he needs an avian vet immediately, because "Force Feeding" a baby bird is not at all recommended, as you can so easily aspirate the formula into his lungs, it happens in an instant and he will die immediately. I'm not exaggerating at all, I saw it happen once and I will never, ever forget it. The baby was around 2+ weeks old, he had a yeast infection in his crop and was being treated by an avian vet for it, but the breeder did not want to "Tube Feed" him (this is the proper way to force feed a bird, but I'll explain later). So this very experienced breeder of decades was taking advantage of what little feeding response this baby had and was pushing too much formula into his mouth when he showed any type of feeding response. The baby suddenly jerked his head upwards, started violently shaking his head back and forth, and then fell over dead. All in about 5 seconds. It happens that quickly, and it happens to very experienced breeders all the time. So you trying to "Force Feed" a 3 week old baby bird is a horrible idea for two reasons, #1 because you can easily kill him, and #2 because you shouldn't have to force feed a 3 week old baby to begin with, he's sick and needs an avian vet.
What you need to do is #1 find a certified avian vet nearest to you, call them and tell them it's an emergency, explain that you have a 3 week old baby IRN that has no feeding response and has not been kept in a brooder at the proper temperature, and if you haven't been feeding the formula at the proper temperature either then tell them that too. That way they'll know what the problem is, that it's an emergency, and they'll know he needs a crop flushing, antifungals, etc. The certified avian vet can also show you how to properly mix his formula at the correct temperature, how to properly feed him safely, get you feeding him on a proper schedule (you should be feeding him overnight until he's close to 5 weeks, every 3-4 hours; at 4 and a half to 5 weeks he can then go 8 hours overnight without eating, but not until at least 4 and a half weeks).
"Dance like nobody's watching..."
