Are there any specific stories in the Satcharita about Udi helping with severe physical weakness or illness?

Yes, Chapter 33 of the Shri Sai Satcharita includes a detailed account in a footnote by R. Hari Sitaram Dixit about a specific healing. It tells the story of Khimji Lalji Joshi's daughter, who was so ill and weak that her limbs were like twigs and she could not walk. After her mother first used Udi from Dixit's house which brought some relief, the family took the girl to Shirdi. As narrated in Chapter 33, a man had to carry her to Sai Baba, who then applied Udi to her and blessed her. Within a few days, she began to walk on her own. When she still had trouble getting up, Baba gave her a fruit to eat, which completely removed that difficulty as well.