It's very encouraging to see a woman asking this question!
Great advice for finding SRO's, especially on the North Side of Chicago: Walk/drive/bike the main thoroughfares of Humboldt Park, and Uptown, and Edgewater, and even Lakeview, even near the Gold Coast. Whenever you see a suspicious looking building with people coming and going, the odds are higher than average that an SRO is inside.
Alternately, some 3 and 4 bedroom apartments, in multi-unit buildings, like your typical three flat, have North Side single professionals looking to rent a room out for near to your price, or your alter ego's. Craigslist has a entire section dedicated to it. Good luck!