A spare room for 3 months: Iván, working in Los Angeles, between homes

After a sudden divorce, I'm temporarily without a home. I work as a teacher's aide in Los Angeles and I have a steady income, but I can't find a place that will let me in without the first month, a deposit and references that I'm in the middle of replacing. I'm looking for a spare room for 3 months. I can pay $270 a month toward bills, I'm tidy and quiet, and I'd be happy to help with cooking, shopping or yard work. I have no pets, I don't smoke, and I have a landlord and employer references. I understand that inviting a stranger into your home is a big step. I'd be glad to meet in a public place first and to share my work details. I never thought I'd be the one asking, and I'm grateful for every bit of support.