It seems counterintuitive that a damp place would help rheumatism. How did staying in the Chavadi actually benefit Amir Shakkar?

This is a profound point addressed in Chapter 22. While the Chavadi's damp and cold conditions were externally contrary to healing rheumatism, the true benefit was spiritual and based on faith. Baba's firm word was the actual 'medicine'. By accepting this difficult instruction, Amir demonstrated his faith. Furthermore, the location was strategically beneficial. Chapter 22 explains that because of the Chavadi's location, Amir received Baba's darshan automatically every morning and evening, and on alternate nights, they were in close proximity, separated only by a curtain. This constant companionship with Sai was a rare blessing that would not have been possible elsewhere.