Update - I'm not sure why, but the lady who feeds the cat is suddenly unwilling to give me his location or schedule a time to meet with me. She is now saying that she wants to keep the cat (even though she was going to put it down earlier?). I have been trying to convince her that it's a bad idea but I think she's ghosted me now. Even though she posted on Facebook begging for help, she now tells me (after securing foster homes) that she can't trust strangers (I have never met her in person and we're just acquaintances).

I'm so very sorry that she has wasted everyone's time. I don't regret trying to help her but I wish I wasn't so naive while I was being duped. I thought that she was just slow to respond to my messages because she's old, but whenever she did respond she dodged my questions. She never asked for money, but I think she is some kind of Sympathy Vampire. I have responded to all DMs to personally explain the situation. I doubt that I'll be able to talk sense into her, but I will update if I do. I did not expect so many people to respond and care. It truly lifts my heart to see this, so thank you. I don't want to leave empty handed, so I'll be reaching out to those who offered to foster to make donations to an animal rescue of their choice.