My budgie's feathers are frayed and skin is dry. What should I do?

Thanks for your question about The Little, and thank you for the photos - they are very helpful! The fraying at the edges of the feathers can be due to him overgrooming, but it may also be due to the feathers being on the body too long (basically a molting interval that is too long). When that happens, the feather can lose some of its strength and can be damaged more easily. These conditions are often a result of stress or nutrition. In budgies, the most common nutritional deficiency that results in this is a lack of vitamin A, which is typically found with seed diets. Ideally, budgies would eat a combination of at least 50% pellets, and the remaining 50% can be made of produce and a small amount of seeds. Vitamin A deficiency can also result in the skin being dry and flaky, and it can also look irritated or red. This may be contributing to the changes that you are seeing on the face and feet as well.
Alternatively, there are some parasites that affect feathers and skin, however if this is the case you have caught it early. I would recommend considering a diet change if applicable, and a wellness evaluation for The Little with your veterinarian to get a thorough physical examination performed.
I hope that gives you a place to start, and thanks for using Petco Pet Education Center, formerly Petcoach!