My project at work is facing an impossible deadline and limited resources, and I feel completely stuck. Is there a lesson from the Satcharita on how to handle such situations?

Yes, there is a very relevant incident in the Shri Sai Satcharita. In Chapter 51, Madhavrao and Dixit faced a seemingly impossible situation. They needed to get to Shirdi, but the second-class train was "packed to capacity," and with "very little time...left," they were gripped by "deep worry" and considered abandoning their journey. This mirrors feeling stuck on a project with no clear way forward. However, their problem was solved in an instant when the train guard, who happened to be an acquaintance of Dixit, suddenly appeared and arranged a comfortable place for them in the first-class carriage. This story shows that even when a situation seems hopeless, help can arrive from an unexpected source if you maintain your faith and purpose.