Very good idea to lower the perches and use a towel, but be sure it's not a terrycloth type so her little toes don't get tangled or stuck in the looping
My biggest concern is that this is a very young bird to be having these symptoms. If you've been feeding her a seed diet you might be looking at hepatic lipidosis (fatty liver disease), internal tumor or even heart issues among other conditions associated with birds on a high fat diet without the means of burning off the calories. In the wild they can eat seeds, although that's not all they eat, and then they burn it off by flying just about constantly every day from dawn 'til dusk. Birds in our homes don't have that opportunity. They're just not going to be chalking up miles of flight time in a house (or cage)
The leg involvement by the way, may not be from injury at all; inability to balance or grasp with one or both legs and feet are also a symptom of health complications due to fatty liver, tumor, etc
Another common condition seen with symptoms like this is hypovitaminosis A (very low vitamin A levels) and again, seed diet
Try this: Offer ½ spoon of all natural, organic baby food (squash, yams, sweet potatoes, mixed vegetables) which many birds take readily; also try some pabulum or baby rice cereal and a few licks of natural (no artificial anything) yogurt. You might see an improvement in a matter of hours and even more if you an convert your little one to a good pellet diet and keep pushing the fresh foods.
Beet greens, Beets, Carrots, Carrot Tops, Chard, Mustard Greens, Spinach, every kind of Pepper (birds are not affected by the capsaicin, the substance that makes peppers hot) just go through your produce aisle and look for dark greens and orange colors.
It’s fun to soak some big leaves of Kale and tuck them all through the cage. The bird may use them as a washcloth while nibbling away.
You can serve veggies raw, chopped, sliced, diced, steamed, boiled, baked or sautéed in a bit of olive oil - be creative.
-- Just because they don’t like something one day doesn’t mean they won’t get used to it at some point in the future.
TRICK: Get a couple jars of human baby food like sweet potatoes, squash, etc.. Many birds will sample this more readily than the actual vegetables at first. As long as you’re sticking with all natural and preferably organic types - it doesn’t matter how they eat it as long as they do.
It's also pretty vital to get her to a vet who knows birds. I'm confident that she doesn't need to live like this and it's something that can be treated.
If you think there's no avian vet nearby, go to Pet Smart. Their Banfield clinic has an avian experienced vet on staff. Call Pet Co and ask for their list of 'bird vets' - they happily share! Call ANY regular vet in town and ask them who they refer bird patients to. Call any breeder in town and ask who they use.
Chances are you have about 2 or 3 of them not too far away.
Good luck and please let me know how you make out ok?