A little bit of an update..
Thank you all for all your suggestions. They were so, so helpful! I'm so thankful for this forum.
My mom contacted the congresswoman in New Mexico, and the lady says this happens a lot. She said to just get as much information as possible, send it in, and if it still comes back rejected, to call the office again, so that' a huge relief.
My mom has contacted the school she might have gone to (grandma is not exactly sure where she went to school.. she only went to 8th grade), and she is getting the sealed baptismal certificate, and this weekend, she is getting her older sister to fill out the affidavit. It sounds like everything will work out!
I'll post when the passport is finally in hand!